Rhode Island will miss out on $300 million in revenue in March and early April as coronavirus delays income tax filings, while shuttered casinos cost state coffers another $1 million per day. Expenses, meantime, are through the roof — the state spent $7 million on ventilators on Monday alone.

Seth Magaziner, the state’s general treasurer, takes comfort in the fact that the federal government should soon cover some coronavirus-related costs, like medical equipment, thanks to legislation President Trump signed last week. But Mr. Magaziner remains worried about the municipal bond market, which state and local governments tap to fund everything from road construction to schools. It could turn messier as government income streams dry up amid quarantines and furloughs, making debt harder to issue. Like many other state finance officials around the country, he’s looking to the Federal Reserve for help.

“Things are very volatile, and we don’t know what the future can bring,” Mr. Magaziner said. “This is a crisis unlike any other that the country has faced in generations.”

State officials, Democratic lawmakers in Washington and advocacy groups are pushing the Fed to step up its efforts to support state and local finances.

The central bank has a century-long practice of staying out of municipal debt markets, in part because doing can involve picking winners. But the coronavirus — and recently passed legislation — is prompting officials to consider what more it can do to help state and local financing. The Fed recently hired Kent Hiteshew, the first director of the Treasury Department’s Office of State and Local Finance, created during the Obama administration. And it’s tapping its 12 regional branches for insight into local conditions.

Officials have already tiptoed into the municipal debt market, accepting those bonds as collateral in a program that provides cheap loans to eligible banks.

But it has yet to make the leap into outright purchases, which its emergency lending powers could allow, even as lawmakers urge it to do more to help states and localities. Buying municipal debt directly could prove practically — and politically — fraught for the central bank.

“It’s not as easy as the Fed going in and saying that they’re going to buy Treasury debt,” said Christopher Brigati, the head of municipal trading at Advisors Asset Management, referring to Fed’s continuing federal government debt purchases.

Whether it’s through the Fed or congressional action, he said, “I want to encourage caution to anyone who is promoting bigger action.”

The problem is that municipal bonds are a relatively tiny market: They total about $3.9 trillion, peanuts compared with the $16.7 trillion Treasury market. And while Treasury bonds are fairly comparable to one another, municipal debt is not.

The bonds fund everything from city parks to airports to state operations, so while they are issued credit ratings, the securities rely on vastly different revenue streams and local conditions. New issuance is subject to unique local restrictions.

Buying big chunks of municipal bonds, as the Fed does in the Treasury market and plans to do in the investment-grade corporate debt market, would be no small task. The securities are not quoted on an exchange, Mr. Brigati said, and a buyer can’t just go to a few dealers to snap them up at scale.

“It’s challenging to source these bonds,” said Vikram Rai, the head of municipal strategy at Citigroup. Mr. Rai also said buying fresh bond issues could come with difficulties.

“It’s a very politically sensitive issue to buy a municipal bond,” he said. He laid out a scenario: If New York issues debt in April, when the Fed is buying, it will be able to price its bonds better. If New Jersey issues in September, when the Fed isn’t, it could find itself at a disadvantage.

Even if the Fed sticks to the market for existing bonds, it risks benefiting a few major bond holders in the space in its quest to snap up a large volume of bonds. Money market mutual funds and exchange-traded funds hold big shares of the market; the investment firm BlackRock calls itself the world’s largest municipal bond manager, while Nuveen and Pimco are also major players.

“Most people don’t know what the Fed is planning,” said Matt Fabian, a partner at Municipal Market Analytics. “There’s a lot of fear on the part of the broker-dealer community that the Fed could mess things up.”

Problems of picking winners and losers and benefiting existing debt holders also plague other Fed emergency lending programs, like corporate bond buying, said Skanda Amarnath, the research director at Employ America, a worker advocacy group.

They shouldn’t be a deciding barrier, he said. And while it will be practically difficult for the Fed to jump headlong into the idiosyncratic municipal market, he said, officials should still work it out.

“It’s a more challenging problem,” said Mr. Amarnath, who has been organizing state treasurers around the cause. “But it’s also a more worthwhile challenge, because there’s more at stake here.”

After the 2008 financial crisis, state and local belt-tightening helped to keep unemployment high for years, serving as a drag on the recovery. And that shock hit much more slowly than the one currently playing out, as coronavirus abruptly chokes off sales taxes from shuttered stores and makes it hard for airport vendors, the source of cash for many airport bonds, to keep up on rent.

“Without timely and strong federal government efforts to support the municipal bond market and compensate for delayed revenues, our state and local governments will be forced to take actions that will exacerbate economic contraction,” the National Association of State Auditors, Comptrollers and Treasurers wrote in a letter to the Fed and Treasury late Tuesday.

The group urged the Fed to set up a temporary “Municipal Securities Purchasing Facility” to calm the market for state and local debt, where new issuance has become difficult. They suggested it should buy existing debt from a variety of issuers with a range of credit ratings, and also recommended that the Fed create a bridge financing program for states and local governments.

William C. Dudley, who was formerly president of the Federal Reserve Bank of New York, is sympathetic to the idea of a municipal bond program of some sort, especially given that “they basically have the blessing of Congress.” But he said officials should stick to investment-grade debt.

“Once you start to get below investment grade, it gets a lot trickier — you’re taking on a lot more risk, and where does it end?” Mr. Dudley said. The central bank might end up with riskier bonds that investors are offloading because the debt is unlikely to be paid back.

Lawmakers are leading the push for Fed action. The new legislation insists that Treasury Secretary Steven Mnuchin push for an emergency lending program related to state and local finance, though it’s fuzzy on the details. Speaker Nancy Pelosi, Democrat of California, has said repeatedly that she’s urged Jerome H. Powell, the Fed chair, to do more to help municipal markets.

Senator Elizabeth Warren, Democrat of Massachusetts, urged Mr. Powell and Mr. Mnuchin in a Tuesday letter to “address the needs of state and municipal governments that face desperate budget shortfalls” and do so “rapidly.”

After legislation passed empowering the Fed with more financial backing for emergency lending, which it would use to snap up the municipal bonds, the market for outstanding local bonds temporarily calmed. But volatility returned on Wednesday.

“Liquidity is the most critical piece right now,” said Deborah Goldberg, who is Massachusetts’ treasurer and who has been pushing the Fed to get involved.

The message from the Fed and the Treasury has been, consistently, that it is a work in progress.

“We’ll be looking at programs for state and local governments,” Mr. Mnuchin said in a CNBC interview Wednesday. “I can assure you, Jay Powell and I are working around the clock at providing liquidity into the economy.”

Mary Williams Walsh and Alan Rappeport contributed reporting.