Congressional leaders and the White House are converging on the need for a new assistance package to try to contain the coronavirus pandemic’s economic devastation, fearful that a $2 trillion bailout law enacted last month will have only a limited effect.

House Democrats are eyeing a package of spending increases that would “easily” cost more than $1 trillion, Speaker Nancy Pelosi (D-Calif.) told lawmakers Monday, according to two officials on the conference call who spoke on the condition of anonymity to discuss it. Democrats are looking to extend unemployment aid and small-business assistance for additional months, as well as authorize another round of direct checks to taxpayers.

Trump has signaled support for some of the ideas that Democrats back, such as expanded help for small-business owners and new bailout checks for households. Republican leaders, meanwhile, have also called for more corporate aid and money to boost the overwhelmed health-care system.

The rushed effort comes as the economy’s fortunes appear tethered to political decisions in Washington. A surge of demand for new Small Business Administration loans has overwhelmed the agency’s system, but the Dow Jones industrial average rose 1,627 points, or 7.7 percent, on Monday amid Wall Street optimism that the pandemic’s domestic spread could be slowing.

Although there is some overlap in the political ideas, significant differences remain, and it’s unclear whether the sides will be able to reach a deal in the coming weeks.

Democrats are pushing for wider-ranging relief measures to help hospitals, health-care workers, farmers, first responders and education programs, among other things. A group of House Democrats led by Rep. Joe Neguse (Colo.) plans to roll out a proposal for $250 billion to assist smaller cities and communities they say have been shortchanged on assistance so far. A number of congressional Republicans have demanded an expansion of a new $349 billion small-business loan program, and Trump has also said that initiative needs more money.

In a sign that lawmakers might be preparing to cut a deal, Pelosi has backed away from some of her recent proposals that Republicans found most objectionable, including a multi-trillion-dollar infrastructure plan. And Senate Majority Leader Mitch McConnell (R-Ky.) has said he believes that Congress will have to act again to address health-care needs, among other things, opening the door to a pact. The two leaders spoke Thursday.

“All I can tell you is, I think we’re going to definitely need” another rescue bill, Senate Minority Leader Charles E. Schumer (D-N.Y.) told reporters Monday. “And I think it’s going to have to be big and bold because the problem is so huge.”

Their work to mobilize new legislation came amid growing signs that the economy is deteriorating much faster than expected and that the initial $2 trillion law is proving insufficient. Former Federal Reserve chair Janet L. Yellen told Pelosi and House Democrats on their conference call that the actual unemployment rate is probably 13 percent, not the 4.4 percent the Bureau of Labor Statistics reported Friday. And Yellen said the number of new jobless claims this week is likely to exceed last week’s 6.6 million, a record, according to Democrats on the call.

Households and businesses are confronting so much turmoil that the new assistance programs are overwhelmed. State unemployment offices have been bombarded with people seeking help.

And Bank of America said Monday that it had received 178,000 applications from firms seeking $32.9 billion in loans as companies clamor to qualify for the $349 billion Small Business Administration program.

Wells Fargo didn’t begin taking applications until Saturday and by Monday morning said it had reached the $10 billion cap it had set for loans under the program. To deal with the crushing demand, the Federal Reserve launched a system for banks to offload these assets so they could originate more loans.

This program is supposed to encourage small businesses to stop laying off employees, after 10 million workers sought unemployment claims late last month. The unemployment rate is expected to surge far beyond 10 percent this spring, and it could stay there into next year.

Goldman Sachs projected that the jump in new spending and sharp drop in tax revenue would push the federal budget deficit to $3.7 trillion in 2020, up from prior estimates of $1 trillion, which many experts said was already too high.

And in his annual letter, JPMorgan Chase chief executive Jamie Dimon predicted that the economic fallout from the pandemic would lead to a “bad recession” and create financial stress rivaling the 2008 crisis.

A new economic package would be the fourth piece of legislation focused on the coronavirus to move through Congress since early March and would aim to extend and bolster programs created in the $2 trillion Cares Act.

The office of Sen. Marco Rubio (R-Fla.) said over the weekend that the lending fund for small businesses — billed as the Paycheck Protection Program — will be depleted before it expires June 30, and “it is clear that Congress will need to appropriate additional money.”

Across the nation, state unemployment offices are staggering under the weight of an unprecedented number of claims, and lawmakers called for expanding the $260 billion program they created last month that offers unemployed workers an additional $600 weekly for four months on top of what their state programs provide.

In a letter to Democratic lawmakers Sunday, Pelosi said, “The acceleration of the coronavirus crisis demands that we continue to legislate.”

“We must double down on the down-payment we made in the Cares Act by passing a Cares 2 package, which will extend and expand this bipartisan legislation to meet the needs of the American people,” she said.

Pelosi and House Democrats last week rolled out a host of more ambitious — and controversial — measures, including new federal workplace safety standards for health-care providers and the infrastructure plan. But Pelosi later told CNBC that those ideas might have to wait.

“While I’m very much in favor of doing some of the things that we need to do to meet the needs — clean water, more broadband and the rest of that — that may have to be for a bill beyond this,” Pelosi said in an interview Friday. “Right now, I think that we have a good model — it was bipartisan, it was signed by the president, but it’s not enough.”

Trump has indicated openness to more money for any number of additional programs.

“If we do more, we’re going to do more. We’re going to try and get directly to people that are hit so hard,” Trump said Sunday. “But we’re going to take care of our workers. We’re going to take care of our citizens. We’re going to take care of our small businesses. We’re going to take care of our large businesses, the airplane industry, the airline industry, a lot of industries that we have that are in trouble.”

A senior administration official, speaking on the condition of anonymity Monday to describe internal deliberations, said the administration was focused on implementing the programs in the new law. Several of these are experiencing rocky rollouts as the Treasury Department and the Internal Revenue Service try to move quickly to push out loans to businesses and send checks of between $500 and $1,200 to individual Americans making $99,000 or less.

But the administration is not ruling out another piece of legislation building on existing programs.

“We clearly hear it from the Hill including [unemployment insurance], payments to individuals, how we do another small-business program, and then lots of conversations about things that were left out or groups of people that were left out,” this official said.

One of the gaps in the Cares Act that lawmakers are looking to fill deals with aid to local governments. While the bill reserved $150 billion for states and local jurisdictions with more than 500,000 residents, smaller cities, towns and counties are not directly eligible.

Neguse — in concert with Democratic Reps. Andy Levin (Mich.), Ben Ray Luján (N.M.) and Tom Malinowski (N.J.) — is proposing to earmark $250 billion for smaller jurisdictions in the next relief bill. Dozens of lawmakers of both parties previously signed letters calling for the fix.

“It does not matter what political affiliation you might have or whether you live in a big city or a small city, the virus at the end of the day is impacting all of us as Americans,” Neguse said Monday. “There are cities across our country, counties, that are going to be in very dire economic circumstances very soon if the Congress does not step up to the plate.”

Sen. Shelley Moore Capito (R-W.Va.), a senior member of the Senate Appropriations Committee, said Monday that lawmakers need to “see what’s working, what’s not working, anybody falling through the cracks.”

As thorny as the policy questions about what would be in the next bill is the question of how to pass it.

The House and Senate are out of session over coronavirus fears, and many lawmakers are not eager to return to the close quarters of the Capitol. But to pass legislation without lawmakers present, leadership in both chambers and of both parties would have to get unanimous agreement from all members of Congress, something that may be difficult to achieve.

Because one House member, Rep. Thomas Massie (R-Ky.), demanded a recorded vote on the Cares Act, leaders scrambled to bring more than half the chamber’s members back to ensure that the law could pass by voice vote. Aides of both parties, who spoke on the condition of anonymity to candidly describe internal thinking, said it is more unlikely that subsequent bills will be able to pass unanimously or on voice votes.

The Senate has tentatively scheduled votes for the week of April 20, and the House leadership has also targeted that week for potential votes. But aides say the legislative schedule is entirely dependent on the course of the pandemic and whether the two chambers and Trump can come together on workable legislation.

Paul Kane contributed to this report.