WALL STREET JOURNAL, KRISTINA PETERSON and ANDREW DUEHREN

Lawmakers released a blitz of competing coronavirus relief proposals Tuesday, reigniting stalled talks as the pandemic surges across the country, but without any clear signs that Democratic and Republican leaders would be able to reach a consensus.

A bipartisan group from the House and Senate unveiled a roughly $900 billion compromise proposal, offering one route between House Democrats’ last $2.4 trillion bill and Senate Republicans’ recent $650 billion proposal. Meanwhile, GOP and Democratic leaders traded new offers, as the looming expiration of some relief measures intensified the pace of year-end legislating.

In another sign of the new urgency, House Speaker Nancy Pelosi (D., Calif.) and Treasury Secretary Steven Mnuchin held their first phone conversation since the election on relief measures and spending legislation.

In an indication that GOP leaders weren’t yet ready to make major concessions, Senate Majority Leader Mitch McConnell (R., Ky.) circulated a new plan to his caucus that he said reflected what President Trump would be willing to sign into law in the waning days of his administration, after speaking with White House officials.

The new Republican offer, similar to the recent Senate GOP bill, includes more funding for small businesses, legal protections for entities operating during the pandemic and a one-month extension of expanded unemployment insurance programs that expire at the end of December. It doesn’t include additional funding for state and local governments, a priority of Democrats. There was no official price tag for the new GOP offer.

“I hope this is something that would be signed into law by the president, be done quickly, deal with the things that we can agree on now,” Mr. McConnell told reporters. He nodded to the possibility of more aid proposals next year under the incoming Democratic administration.

On Monday night, Mrs. Pelosi and Senate Minority Leader Chuck Schumer (D., N.Y.) sent a letter to Mr. McConnell outlining Democratic leaders’ latest offer.

Mr. Schumer declined to give details of the proposal, saying that it was intended to revive bipartisan negotiations. He said the bipartisan proposal outlined earlier Tuesday was a positive step toward compromise and he criticized Mr. McConnell for pursuing another Republican proposal, which Democrats have called insufficient.

“The obvious fact of the matter is the biggest impediment to getting an agreement is the Republican leader refusing to negotiate in a bipartisan way,” Mr. Schumer said.

Democratic President-elect Joe Biden again urged Congress to reach a virus-relief agreement before the end of the year. He said that “any package passed in the lame duck session is likely to be — at best — just a start.”

The Republican proposal would also repurpose money from Federal Reserve lending facilities, which Mr. Mnuchin is letting expire at the end of the year. Unemployment programs that expand eligibility for receiving jobless benefits and extend the timeline for receiving them would be continued for another month, into January. People who already participate in the programs could draw benefits for two additional months.

Under the GOP plan, businesses could receive a second loan through the Paycheck Protection Program, and schools and colleges would receive more than $100 billion in aid, while $31 billion would go toward vaccine development and distribution.

Mr. Mnuchin told a Senate hearing Tuesday that the White House and GOP leaders were interested in further relief, but didn’t provide a target. He said that he had spoken in recent days with Mr. Trump, Mr. McConnell, House Minority Leader Kevin McCarthy (R., Calif.) and White House chief of staff Mark Meadows about aid.

“We all believe there should be targeted fiscal response,” he said.

Lawmakers are also working to finalize a full-year spending bill before the government’s current funding expires at 12:01 a.m. on Dec. 12. Last week, Senate Republicans and House Democrats agreed on how to divvy up the $1.4 trillion pot of money across the 12 spending bills that fund the government. Mrs. Pelosi mapped out the progress in a phone call with Mr. Mnuchin Tuesday.

“I relayed my hope that the Administration would support this bipartisan path,” she said in a statement after the call. She said Mr. Mnuchin told her that he would review both the Democratic leaders’ coronavirus relief proposal and the new bipartisan offer.

Mr. Mnuchin said Tuesday that Mr. Trump “wants to make sure there’s not a shutdown.”

Earlier Tuesday, a group of lawmakers from both the House and Senate unveiled a $908 billion aid proposal, highlighting the growing pressure from rank-and-file lawmakers to pass additional coronavirus aid.

“We simply can’t leave town and leave anyone in the cold, without getting something done immediately,” said Rep. Josh Gottheimer (D., N.J.), co-chair of the bipartisan Problem Solvers Caucus, a group of 50 lawmakers, which has endorsed the proposal.

The proposal, which would run through March 2021, includes some measures sought by both parties. Lawmakers included $160 billion in state and local funding, long one of the biggest sticking points in talks among congressional leaders.

The bipartisan proposal also gives a nod to one of Republicans’ top priorities: legal protections for businesses and other entities. Lawmakers said they would provide a short-term suspension of liability lawsuits related to Covid-19 at the state or federal level, giving states time to put in place their own protections.

The proposal also includes $288 billion for small-business relief, including PPP, $16 billion for the distribution of a coronavirus vaccine, $82 billion for schools, $25 billion for rental assistance and $180 billion for additional unemployment insurance, including $300 a week through March, aides said. In addition, the plan would give $17 billion to airlines.

Republicans said they had worked to keep the overall cost in check, after GOP lawmakers balked at the size of Democrats’ previous proposals.

“We’ve been very careful. This is not a $1.8 trillion stimulus bill,” Sen. Mitt Romney (R., Utah), part of the bipartisan group, said Tuesday. “This is a relief measure—half that amount.”

Rep. Abigail Spanberger (D., Va.) said the group worked intently on the proposal, even holding a morning call on Thanksgiving Day, to hash out the agreement.

“To be clear, no one is saying it’s perfect,” she said. “It is a deal that people can agree upon and most importantly it would deliver help to the people who need it.”

If leaders are able to reach an agreement on Covid-19 relief, it is likely to be wrapped into the spending bill, rather than passed on its own, Mr. McConnell said.