The package under discussion was expected to include, along with direct checks, $300 a week in enhanced unemployment insurance, funding for vaccine distribution, schools, small businesses and health-care providers, and other relief measures. Its size, at just under $900 billion, marked a compromise between the two parties’ stances: more than the roughly $500 billion Republicans had backed and less than the $2.4 trillion bill Democrats passed in the House earlier this year.

“We’re still talking and I think we’re going to get there,” Senate Majority Leader Mitch McConnell said Wednesday afternoon.

Although much of the agreement mirrored a proposal crafted by a bipartisan group, congressional leaders were still haggling over other elements of the sprawling package they hoped to pass later in the week. Lawmakers expect to attach the aid bill to a full-year spending bill needed to keep the government running after its current funding expires at 12:01 a.m. Saturday.

The aid package under discussion on Wednesday was expected to exclude the two thorniest issues: funding for state and local governments and liability protections for businesses and other entities operating during the pandemic, according to lawmakers.

But congressional leaders were expected to add a second round of direct checks, likely of a smaller size than the initial round, which provided $1,200 to individuals and $500 per dependent earlier this year. Senate Majority Whip John Thune (R., S.D.) told reporters Wednesday he expected the checks would be in the $600 to $700 range per individual.

Congressional aides noted that the negotiations were continuing and no final agreement had been reached.

House Speaker

Nancy Pelosi

(D., Calif.), Senate Minority Leader

Chuck Schumer

(D., N.Y.), House Minority Leader

Kevin McCarthy

(R., Calif.) and Mr. McConnell met multiple times on Tuesday, talking late into the night, and continued their negotiations Wednesday.

Treasury Secretary

Steven Mnuchin

joined the two meetings by phone. The

Trump

administration has previously proposed sending $600 checks.

“We made major headway toward hammering out a targeted pandemic relief package that would be able to pass both chambers with bipartisan majorities,” Mr. McConnell said on the Senate floor on Wednesday. Mr. McConnell had called last week for leaving behind the two most contentious issues and proceeding with a narrower deal.

“We are close to an agreement. It’s not a done deal yet, but we are very close,” Mr. Schumer said on the Senate floor. “For Democrats, this has always been about getting the American people the relief they need at a time of an acute national crisis.”

Mr. Schumer said that Democrats would seek further aid next year with the new administration. Democrats said Tuesday that states and local governments would need more help next year to keep paying the salaries of police officers and teachers, among other effects of the budget shortfalls.

President-elect

Joe Biden

applauded the emerging deal on Wednesday, but reiterated that he viewed it as a down payment on additional aid that would be required next year.

“We’re going to have to revisit that. Many state and local governments, including mine, are still in really tough shape,” said

Rep. Josh Gottheimer

(D., N.J.), co-chairman of the Problem Solvers Caucus, a bipartisan group of 50 House lawmakers. Leaders of the group were involved in crafting a bipartisan proposal that congressional leaders used as the basis of their negotiations. But “after nine months of absolute stalemate, when people are hurting…something is better than nothing,” he said.

Lawmakers have only a few days to complete the aid package and spending bill and approve them in both chambers. Lawmakers and aides said they were making good progress completing the full-year spending bill. The package is expected to pass the House first, then arrive in the Senate.

Congressional leaders hope to keep the package’s total cost around $900 billion and were working Wednesday to see how its components fit together. They were discussing reducing the period of enhanced unemployment insurance to 10 weeks from 16 weeks to pay for the second round of direct checks, according to people familiar with the discussions. And some proponents of the checks were likely to object to a sum less than the $1200.

“That’s progress. It’s not where I would like it to be or I think it should be,” said Sen.

Josh Hawley

(R., Mo.), who has been pushing with

Sen. Bernie Sanders

(I., Vt.) to add another round of $1,200 checks.

For weeks, Democrats had sought to include funding for state and local governments facing budget shortfalls after months of pandemic-related closures and economic fallout. Republicans had resisted, saying not all states needed an infusion of aid.

Meanwhile, Mr. McConnell and GOP lawmakers had pushed for legal protections for businesses, schools and nonprofits operating during the pandemic, saying that liability limits would help the economy fully reopen. Democrats said they were concerned that the GOP proposals went too far and wouldn’t incentivize businesses to take necessary precautions to protect workers.

The top four congressional leaders began negotiating in person on Tuesday, one day after a bipartisan group released a $748 billion bill that fleshed out their framework for an emergency relief deal through March. The largely rank-and-file group, which had initially proposed a $908 billion offer, spun off state and local government funding and liability protections into a separate bill.

The core $748 billion bill, which the bipartisan group unanimously supported, includes an additional $300 a week to state unemployment insurance recipients for 16 weeks, $300 billion for small businesses, including another round of the Paycheck Protection Program, $35 billion for health-care providers and $82 billion for schools.

Mr. Schumer said Tuesday that he hoped to beef up the proposal’s $6 billion for Covid-19 vaccine distribution, as the first doses of a vaccine were administered this week. The bipartisan coalition also included roughly $10 billion for testing and tracing of the coronavirus.

Democratic and GOP leaders have said Congress shouldn’t adjourn until they have passed a coronavirus aid package. Mr. McConnell has said that lawmakers would be able to revisit the issues of state and local funding and liability protections next year, given Mr. Biden’s intent to pass more aid.

President Trump, who has called for a new round of checks to most Americans, is waiting to see what a possible deal looks like, White House press secretary

Kayleigh McEnany

said Tuesday.

“He’s said that he would really like to see those stimulus checks in there, but his priority at the end of the day is getting relief to the American people,” she said. “We’re hopeful there will be some sort of agreement.”

Write to Kristina Peterson at kristina.peterson@wsj.com and Andrew Duehren at andrew.duehren@wsj.com