WASHINGTON, D.C. —One of the most celebrated pieces of the massive $2.3 trillion coronavirus relief package that’s just over 2 weeks old is on life support with no deal in sight to resuscitate it.

According to the latest projections, the Small Business Administration’s $349 billion appropriation for loans to eligible firms, which are forgiven if they use the money to keep workers on their payrolls, will run out of money by Friday.

The White House and Republican leaders want to boost that amount to $600 billion to give more small businesses an opportunity to get relief before the funding dries up.

But Democrats see an opportunity to fix flaws in the so-called Paycheck Protection Program that have become apparent since its April 3 rollout, as well as tack on hundreds of billions of dollars more for state and local governments and hospitals that are still bleeding cash.

Their case has been bolstered by groups like the National Restaurant Association, which wants changes to the SBA program in addition to more funding, and the U.S. Chamber of Commerce, which supports a related small-business loan fund that’s also short of cash.

And on Saturday, the bipartisan leadership of the National Governors Association — including Democratic Gov. Andrew M. Cuomo, from Senate Minority Leader Charles E. Schumer’s home state of New York — asked for as much as $500 billion in federal aid for states.

That’s far in excess even of what Schumer and Speaker Nancy Pelosi, D-Calif., have proposed so far, and doesn’t include additional direct aid to local jurisdictions. But Cuomo and Maryland Gov. Larry Hogan, a Republican, said in a statement that without the added funds, “states will have to confront the prospect of significant reductions to critically important services all across this country.”

Talks between top Democratic leaders and their favorite GOP negotiating partner, Treasury Secretary Steven Mnuchin, got off to a seemingly positive start Friday. Schumer said a phone call with Mnuchin had been “constructive” and added that a bipartisan deal was possible “early next week.”

But sources familiar with the discussions say not much progress was made over the weekend, due in part to the Easter holiday.

After it became clear that the small-business lending fund had burned through roughly half its money in one week, with the daily “burn rate” only ramping up, GOP leaders on Saturday appeared to undercut attempts at a compromise. They released a blistering statement calling on Democrats to drop their objections to their two-page bill that would provide the additional money and nothing else.

“All we want to do is put more money into a popular job-saving policy which both parties designed together,” Senate Majority Leader Mitch McConnell of Kentucky and House Minority Leader Kevin McCarthy of California said in a joint statement. “American workers are in crisis. Nobody except Washington Democrats seems to be unclear on this fact or confused about the urgency.”

Senate Finance Chairman Charles E. Grassley, R-Iowa, piled on in a television appearance Saturday, telling Fox News’ Neil Cavuto that with the loan program dwindling, Democratic leaders “don’t have a leg to stand on.”

The initial $349 billion in the huge relief bill President Donald Trump signed March 27 was intended to cover payroll, rent, mortgage interest and utility costs for eligible firms that apply before June 30.

Loan amounts can be for up to 250 percent of monthly payroll expenses, and eight weeks’ worth of debt is forgiven if loan recipients use the money for intended purposes and spend at least three-fourths of the money on payroll.

On Friday, White House National Economic Council Director Larry Kudlow projected that the small-business loan program will run out of money April 17. “That’s why we would like the Congress to help us with an additional $250 billion,” he told Fox Business’ Lou Dobbs.

Kudlow said at the time the government had approved 661,000 small-business loans, valued at $168 billion.

Others said the money could run out before Friday.

A Senate Small Business and Entrepreneurship Committee aide said the panel’s model shows the average daily amount of loans approved continues to rise. “The committee is not publicizing a date because the situation is fluid,” the aide said.

But if the SBA burned through about half its money by the end of the week, a rising burn rate implies it could end up completely dry even before April 17.

To Democrats, the answer is simple: a negotiated solution that pumps more money into the SBA loan fund, but also moves in their direction in three chief areas.

According to Pelosi spokesman Drew Hammill, the speaker told Mnuchin in their call Friday that the SBA cash infusion “must not solidify the disparity in access to capital faced by many small businesses in underserved areas.” She also said hospitals and states and localities need more money.

The Democrats’ alternative to the Republicans’ two-page bill adding $251 billion to the small-business fund would set aside only half of the money for the existing program.

Instead, an extra $125 billion would be allocated differently, including $60 billion for smaller financial institutions that specialize in lending to minority-owned businesses and those without established relationships with the big banks that dominate the existing SBA program.

An additional $50 billion would replenish funding for disaster loans of up to $2 million each available to businesses for “economic injury.” According to Democrats, the $50 billion appropriation would subsidize some $300 billion in additional loans.

The disaster loan program got $1 billion in an earlier aid package to support some $7 billion in loans, but the program has over $370 billion worth of demand, according to Sen. Benjamin L. Cardin, D-Md., the ranking member on the Small Business Committee.

The U.S. Chamber of Commerce said in a letter to lawmakers Friday that far from the maximum $2 million, or even the average request of $200,000, the best the SBA has been able to do is ration the disaster loans down to $15,000 per applicant due to lack of money.

“Such limited economic relief will be insufficient for a great many main street employers,” wrote Chamber president Suzanne Clark.

The Democratic proposal would also pump in an extra $15 billion to provide $10,000 cash advances on their disaster loans that wouldn’t have to be repaid. The most recent aid package had $10 billion for that purpose, but that money is also already gone, Cardin said on the Senate floor April 9.

On state and local aid, it seems unlikely governors will get the full $500 billion they want. The Democrats proposed another $150 billion, which would double amounts already provided and fix problems with the earlier funding round cited by critics.

Cuomo, for instance, blasted the estimated $4 billion allocated to New York state under the initial formula. The Democrats’ new proposal would add perhaps $10 billion more for New York at the state level, given a new set-aside according to each state’s share of the nationwide COVID-19 infection rate, plus billions of dollars more for municipalities.

The measure would also allocate substantially more for the District of Columbia than the first $150 billion funding round did last month, addressing another Democratic demand that the nation’s capital get at least as much as states with small populations like Wyoming and Idaho.

Finally, the entire $300 billion provided in both rounds of aid would be fungible for states and localities, meaning they could use it to fill steep budget shortfalls caused by the pandemic and associated economic shutdown. The first batch of aid stipulated that it could only be used for costs associated directly with COVID-19.

Hospitals in New York and elsewhere have complained of getting shortchanged in the Department of Health and Human Services’ distribution of the first $30 billion in direct aid appropriated in last month’s massive aid package. The agency says the remaining $70 billion is forthcoming, including more for rural and underserved hospitals, but Democrats want an additional $100 billion.

McConnell could opt to try to pass the clean SBA cash infusion Monday when the Senate convenes for a pro forma session. But Democrats’ objection on April 9 diminishes the odds he’d try again, a move Schumer and others derided as “a political stunt.”

Similarly, there’s no current path to passing the Democrats’ proposal without changes.

And even if there’s a midweek deal, the White House and congressional leadership still would have to confront Rep. Thomas Massie, R-Ky., who has made clear he’ll require lawmakers to return to Washington to vote on any proposal, unless remote voting procedures are instituted.

All of this drama means some small businesses that have not been approved yet for assistance may have to wait longer for relief.