“The White House says they don’t support that, but we do,” she said.

Congress initially allocated $349 billion to the program, which is intended both to help small businesses and to encourage them to continue paying and employing their workers. Under its terms, the loans provided are to be forgiven for businesses that maintain their staffing levels and use the money to cover the cost of their payrolls. Mr. Mnuchin has repeatedly called on Congress to approve the request, which was formally sent to Capitol Hill as $251 billion on Tuesday evening.

“I want to assure all small businesses out there we will not run out of money,” Mr. Mnuchin said Wednesday on CNBC, adding that President Trump asked him to return to Congress with the request. “We hope they pass this tomorrow and Friday, and we want to assure everybody if you don’t get a loan this week you’ll get a loan next week or the following week. The money will be there.”

He did not address the demands raised by Democrats. Ms. Pelosi said she hoped an agreement could be reached with the administration.

Since it began Friday, the Paycheck Protection Program has had an exceedingly rocky start. It has stretched the limits of the Small Business Administration, the federal agency in charge of coordinating the loans. Small-business owners, bankers and other participants have voiced concerns that very little of the billions disbursed have actually reached companies in need of the money, and lawmakers are feverishly working to remedy the initial glitches.

But the rush to replenish it, lawmakers said, served as an indication of just how many small businesses wanted to take advantage of the program. Mr. Mnuchin told Democratic lawmakers on a conference call on Wednesday that the program had approved more than $98 billion in loans to more than 3,600 different lenders, according to an official familiar with the call, and the Treasury Department issued additional guidelines on Wednesday to help ease the application process.

He also told lawmakers that the first round of direct cash payments to American families were set to begin next week, according to the official.

Even the perception that there might not be enough money to distribute to the thousands of businesses seeking aid could cripple companies, as a typical small business cannot survive less than a month without incoming revenue, according to research by the JPMorgan Chase Institute.