WASHINGTON — A coalition of 10 Republican senators took a stimulus counterproposal to the White House on Monday evening, urging President Biden to scale back his ambitions for a sweeping $1.9 trillion pandemic aid package in favor of a plan less than one-third the size that they argued could garner the bipartisan consensus the new president has said he is seeking.

Their outline, which came as Democrats prepared to push forward on Mr. Biden’s plan with or without Republican backing, amounted to a test of whether the president would opt to pursue a scaled-back measure that could fulfill his pledge to foster broad compromise, or use his majority in Congress to reach for a more robust relief effort enacted over stiff Republican opposition.

Mr. Biden appeared eager to signal an openness to negotiating, telling Senator Susan Collins, Republican of Maine and the leader of the group, that he was “anxious” to hear what the senators had to say as they chatted in the Oval Office before the meeting began, and spending two hours behind closed doors in what both sides described as a cordial and productive session.

“I wouldn’t say that we came together on a package tonight,” Ms. Collins told reporters as she left. But she called the discussion “excellent” and said Mr. Biden and the senators had agreed to continue their talks. And she said Republicans appreciated that Mr. Biden had devoted his first Oval Office meeting as president to “a frank and very useful discussion” with them.

“All of us are concerned about struggling families, teetering small businesses, an overwhelmed health care system, getting vaccines out and into people’s arms, and strengthening our economy and addressing the public health crisis that we face,” Ms. Collins said.

Even so, Mr. Biden’s advisers have made clear that the president has little enthusiasm for significantly cutting back on the rescue measure he has proposed. And there was scant evidence, for now, that any Democrats were seriously considering embracing a proposal as limited as the one the Republicans have laid out.

“The risk is not that it is too big, this package,” Jen Psaki, the White House press secretary, said before the meeting. “The risk is that it is too small. That remains his view.”

Republicans outlined the plan as the Congressional Budget Office projected that the American economy would return to its pre-pandemic size by the middle of this year, even if Congress did not approve any more federal aid for the recovery, but that it would be years before everyone thrown off the job by the pandemic would be able to return to work.

The rosier-than-expected projections were likely to inject even more debate into the discussions over the stimulus measure, emboldening Republicans who have pushed Mr. Biden to scale back his plan. But they also indicated that there was little risk that another substantial package of federal aid could “overheat” the economy, and reflected the prolonged difficulties of shaking off the virus and returning to full levels of economic activity.

The Republicans’ $618 billion proposal would include many of the same elements as Mr. Biden’s plan, with $160 billion for vaccine distribution and development, coronavirus testing and the production of personal protective equipment; $20 billion to help schools reopen; more relief for small businesses; and additional aid to individuals. But it differs in ways large and small, omitting a federal minimum wage increase or direct aid to states and cities.

It would slash the direct payments to Americans, providing $1,000 instead of $1,400 and limiting them to the lowest income earners, excluding individuals who earned more than $50,000. It would also pare back federal jobless aid, which is set to lapse in March, setting weekly payments at $300 through June instead of $400 through September.

On Capitol Hill, top Democrats said they were worried a smaller package would not adequately meet the needs of struggling Americans.

“This proposal is an insult to the millions of workers and families struggling to survive this crisis,” Senator Ron Wyden, Democrat of Oregon and the incoming Finance Committee chairman, said of the Republican plan. “A bill that sets up yet another cliff for jobless workers in a few short months is a nonstarter.”

Hours before Mr. Biden sat down with the Republicans, Democratic leaders began laying the groundwork to move forward on their own, if necessary, with the president’s $1.9 trillion plan through a process known as budget reconciliation, which would allow it to bypass any Republican filibuster with a mere majority vote.

Speaker Nancy Pelosi and Senator Chuck Schumer of New York, the majority leader, filed a joint budget resolution to begin the process, with plans for votes in the Senate by week’s end.

“The smartest thing we can do is act big, according to Treasury Secretary Yellen,” Mr. Schumer said. “That is what the Senate is going to do: Act big.”

Mr. Schumer said he welcomed Republican input on the package, but warned against Ms. Collins’s approach.

“The only thing we cannot accept is a package that is too small or too narrow to pull our country out of this emergency,” Mr. Schumer said.

Members of the Republican group argued it was incumbent upon Mr. Biden to move forward in a bipartisan manner after pledging in his inaugural speech to unify the country. The coalition’s size was meant to demonstrate that a bipartisan path was still possible for a stimulus measure. Ten Republicans could band together with the Senate’s 50 Democrats to defeat a filibuster, which requires 60 votes to overcome.

Senator Todd Young, Republican of Indiana, described the plan as a way to “rein in” Mr. Biden’s proposal, which some Republicans on Capitol Hill are deriding as a “bailout” of states run by Democrats.

Publicly and privately, Mr. Biden’s aides say he is open to negotiating details of the final relief bill but is not interested in significantly scaling it back. Aides have expressed openness in particular to reducing — but not eliminating — the amount of aid to local and state governments, and to limiting the cost of direct payments to individuals by tightening income thresholds for eligibility to receive the payments.

But officials remain adamant that significantly more aid is necessary for unemployed workers, impoverished families struggling to pay for food and housing, and schools seeking the resources to reopen safely. In recent days, administration officials have sent detailed breakdowns of schools’ needs for personal protective equipment and other purchases considered crucial for reopening, in order to make the case for Mr. Biden’s proposed spending.

The Federal Reserve chair, Jerome H. Powell, warned last week that the economy was “a long way from a full recovery” with millions still out of work and many small businesses facing pressure.

Budget officials said the rebound in growth and employment could be significantly accelerated if public health authorities were able to more rapidly deploy coronavirus vaccines across the population.

As it stands, the budget office sees little evidence of growth running hot enough in the years to come to spur a rapid increase in inflation. It forecast inflation levels below the Federal Reserve’s target of 2 percent for years to come, even with the Fed holding interest rates near zero.

Other independent forecasts, including one from the Brookings Institution last week, have projected that another dose of economic aid — like the $1.9 trillion package Mr. Biden has proposed — would help the economy grow more rapidly, topping its pre-pandemic path by year’s end.

Mr. Biden called the meeting after receiving a letter from the senators on Sunday requesting a chance to lay out their plan. Mr. Biden, who worked with Ms. Collins when he served in the Senate, called the Maine Republican and invited her and the other signers to the White House. He also spoke with Ms. Pelosi and Mr. Schumer.

As members of Congress strategize about different paths to pass Mr. Biden’s proposal, much attention has focused on centrist Democrats, whose votes will most likely be needed to advance the plan in a 50-50 Senate. Last week, Senator Joe Manchin III of West Virginia — who is the most conservative Senate Democrat, has raised concerns about direct payments to higher-income Americans and said he wanted to find a “bipartisan path forward” — told reporters that his goal was to make sure Mr. Biden would be a success.

“We’re going to make Joe Biden successful,” he said without elaborating. “I’m for my friend Joe Biden being a successful president.”

But later that evening, a White House official had to call Mr. Manchin to smooth things over, according to a person familiar with the conversation, after he made it clear that he was annoyed that Vice President Kamala Harris had done a local television interview in his state. Mr. Manchin apparently saw the interview as a way to pressure him to support the stimulus plan without talking to him directly.

Also in West Virginia, Mr. Biden received an important boost on Monday: Gov. Jim Justice, a close ally of former President Donald J. Trump, said he supported a bigger relief package than the one that the Republicans senators were proposing.

“If we actually throw away some money right now, so what?” said Mr. Justice, a former Democrat who switched parties to support Mr. Trump in 2017, told CNN.

Senator Jon Tester, a Democrat from conservative Montana, told reporters on Capitol Hill on Monday that he believed a stimulus package needed to be “big enough to get the job done.”

“I think there’s opportunity there to make some changes to it,” Mr. Tester said. “If push comes to shove, if it doesn’t get changed, I’ll vote for the $1.9” trillion, he said.

Congress approved more than $4 trillion through a series of bills last year to address the coronavirus crisis and its economic fallout. Most recently, in December, lawmakers passed a $900 billion stimulus plan that included $600 direct checks to many Americans.

Katie Rogers contributed reporting.