The stimulus law provided $349 billion for the initiative, known as the paycheck protection program. Larry Kudlow, the top White House economic adviser, said Tuesday morning that the program had funded 178,000 loans at a value of $50 billion since it opened applications on Friday.

But the debut has been plagued with problems, with lenders and potential borrowers alike encountering difficulties navigating it. It has stretched the limits of the Small Business Administration, which typically backs $30 billion of small-business loans in a boom year — about the same amount banks are now seeking for their customers in a day. Small-business owners, bankers and other participants have said that very little of the billions disbursed have actually reached companies in need of the money, which are desperate for it.

Under the terms of the program, businesses that maintain their staffing levels and use the bulk of the money to cover payroll costs will not need to repay their loans.

Many economists warned lawmakers — before, during and after the debate on the $2 trillion law — that small businesses would need significantly more help from the government in the face of an outbreak that has brought entire sectors of the economy, including dining and hospitality, to a halt.

Any delay in availability of funds for business owners, or even the perception that there might not be enough money to go around, could cripple companies and potentially throw workers into unemployment. The typical small business cannot survive less than a month without incoming revenue, according to research by the JPMorgan Chase Institute.

“It is quickly becoming clear that Congress will need to provide more funding or this crucial program may run dry,” Mr. McConnell said. “That cannot happen. Nearly 10 million Americans filed for unemployment in just the last two weeks. This is already a record-shattering tragedy, and every day counts.”

With lawmakers scheduled to remain in their districts and home states until at least April 20, approval of such funding would require unanimous agreement between both parties in both chambers.