Congressional leaders and the White House moved toward agreement late Saturday on a potential $500 billion package of funding for additional small-business loans and money for hospitals and virus testing, sources familiar with the talks said.

Negotiators were still going back and forth and hadn’t yet locked down an agreement, in part because Democrats continued to press for up to $150 billion in additional aid to state and local governments experiencing steep budget shortfalls as a result of the economic shutdown.

Republicans strongly resisted more for states and localities on top of the $150 billion already included in last month’s roughly $2 trillion aid package, however, according to sources who weren’t authorized to discuss ongoing negotiations.

Nonetheless, Speaker Nancy Pelosi, D-Calif., said in a pre-taped interview set to air Sunday on ABC’s “This Week” that a deal was near. “We’re close … we have common ground,” she said, without going into specifics. “I think we’re very close to agreement.”

Even absent the state and local aid, the package under consideration could be roughly double the $251 billion in additional funding for a small-business loan program, aimed at avoiding layoffs, that the White House requested earlier this month.

The initial $349 billion allocation for the so-called Paycheck Protection Program was used up within two weeks amid enormous demand. Negotiators are looking at substantially increasing the size of the next round of PPP funds well beyond the $250 billion previously discussed, sources said, in part because officials have determined that amount might not last more than a week or so.

While people familiar with the talks cautioned that everything was still in flux, the parameters of a potential agreement could include about $400 billion in small-business funds overall, with most of the additional funds set aside for the PPP. The remainder would be so-called Economic Injury Disaster Loans and grants, which have also proven highly popular with affected companies.

One of the provisions under consideration is to set aside some of the funds for community banks and smaller lenders with the aim of reaching small businesses that have not been able to access the first tranche of assistance.

Negotiators were also discussing about $100 billion in additional aid to hospitals and other health care providers, and to support widespread testing and other initiatives to help states reopen their economies when it is safe to do so. But that number was still fluid and subject to change.

The bulk of the funds would go to health care providers with a smaller chunk for virus testing, development of a vaccine and other initiatives. Senate Minority Leader Charles E. Schumer, D-N.Y., has been pushing for $30 billion in an eventual agreement to ensure widespread availability of COVID-19 tests.

In a “Dear Colleague” letter released Saturday evening, Pelosi suggested more funding for personal protective equipment like face masks and gloves and “more support for our workers on the frontline” remained on the table.

Pelosi characterized the emerging bill as a bridge of sorts to a larger aid package that could be considered next month. Democrats have said that measure will include, among other provisions, an extension of expanded jobless benefits and another round of tax rebates.

“I am pleased to report that we have been engaged in bipartisan negotiations on our interim legislation and our progress is encouraging,” wrote Pelosi, who’s been leading the talks with Schumer and Treasury Secretary Steven Mnuchin.

The package is not expected to include a 15 percent increase in maximum benefits for the Supplemental Nutrition Assistance Program that Democrats sought.

Democrats also wanted to include a waiver of a 20 percent state match requirement for election security grants, which is also not expected to end up in the interim bill.

The Senate comes in for a pro forma session at 2 p.m. Monday, at which time if a deal is reached it could come up for passage by unanimous consent. The House has a similar session at 10:30 a.m. Tuesday.

A unanimous consent request could be blocked by any one lawmaker in either chamber, however.