As the legislation is currently written, Mr. Mnuchin would not have to disclose the recipients until six months after the loans were disbursed. Some Democrats also said the legislation as written could allow Mr. Trump’s real estate empire to take advantage of the federal aid, and after years of being stonewalled by the White House on oversight requests, they worried that the administration would refuse attempts by Congress to keep tabs on the program.

At the White House on Monday evening, Mr. Trump only fueled Democrats’ concerns when he brushed aside questions about who would oversee the funds, saying: “Look, I’ll be the oversight. I’ll be the oversight.”

“It would be very foolish if they didn’t make a deal,” Mr. Trump said, accusing Democrats of “asking for things that bear no relationship to what we’re talking about.”

Mr. Mnuchin has argued that bolstering the newly created fund would allow the Federal Reserve to effectively inject $4 trillion into the economy, and that he needs the authority to swiftly support companies and industries that are on the brink of collapse as large sections of the economy shut down. On Monday, he pushed back against the suggestion that the money would become a piggy bank for Mr. Trump’s allies.

“I’ve heard people refer to this as a slush fund,” Mr. Mnuchin said on the Fox Business Network. “It’s not a slush fund.”

Hanging over the debate for senators in both parties were unpleasant memories of the stinging backlash from the 2008 and 2009 economic recovery programs that were later condemned on the right and left for helping Wall Street at the expense of working Americans. Democrats said they would not be bullied into supporting the stimulus package by Mr. McConnell, despite evident political risks as they face a battle for control of the Senate in November.

“The fact is, we need to learn from 10 years ago,” said Senator Sherrod Brown, Democrat of Ohio. “If we’re going to do relief package, the money needs to go in the pockets of workers.”