It remains unclear how the Treasury Department or the Small Business Administration will pursue investigations of loans that were given to borrowers that should not have been eligible. For the loans to be fully forgiven, businesses are required to demonstrate to banks that they met requirements to maintain staffing levels for eight weeks. The documentation that they provide could be subject to audits by the Internal Revenue Service.

The guidance released on Thursday said borrowers that repay loans in full by May 7 will be deemed by the Small Business Administration to have made their certifications in good faith, leaving them in good standing with the government.

At least four public companies, Shake Shack, Kura Sushi USA, Ruth’s Hospitality Group and ItWorks, have said that they have already given back the funds from the Paycheck Protection Program. Sweetgreen, which is privately held, also said it had returned its $10 million loan.

The Fed, which is infusing even more money into the financial system through its various facilities, said on Thursday that it would publicly disclose the names of companies that benefited from several of its lending programs.

The Fed said it would also release the amount borrowed and interest rate charged as well as overall costs, revenues and fees on programs backed by Congress’s recent appropriations. The Fed Board will publish program reports on its website at least every 30 days, without blacking out the information.

Congress has handed the Treasury Department $454 billion to support Fed lending facilities, which are meant to keep credit flowing through the financial system. Using that layer of taxpayer insurance, the Fed has announced programs that are meant to help midsize businesses, state governments, and large corporations.

The Fed chair, Jerome H. Powell, and Mr. Mnuchin are required to regularly report to Congress on the programs, but it was unclear how much detail they would disclose publicly and in real time.