If it denies the request, the agency risks running afoul of the president, who appoints its leader; but if it accommodates the Trumps, the agency is likely to draw fire from critics.

The Trump Organization was barred by Congress from seeking relief from the $500 billion rescue fund being administered by the Treasury Department, and a Trump Organization executive said on Tuesday that the company had decided not to apply for a federal loan through the Small Business Administration. The company argues that it is seeking only temporary relief from the G.S.A. while the hotel industry globally copes with an extraordinary drop in business.

Along with the broader hospitality industry, the company is expected to take a significant hit from the economic shutdown. The Trump Organization has temporarily closed its hotel overlooking the Las Vegas Strip, cut staff and services at its hotel in New York, and effectively closed its golf clubs in New Jersey and Florida. It also shuttered the Mar-a-Lago club in Florida, which at this time of year would ordinarily be acting as the “winter White House,” as the president refers to it. The Washington hotel’s bar, restaurant and spa are closed, but it is still accepting reservations.

The request to the G.S.A. is one of a number of attempts by the Trump Organization to get breathing room from its lenders and other financial partners.

The company has been talking with Deutsche Bank, the president’s largest creditor, about the possibility of postponing payments on its loans from the bank.

Mr. Trump owes Deutsche Bank more than $300 million on loans connected to the Washington hotel, his Doral golf resort in Florida and a skyscraper in downtown Chicago. The Trump Organization has a small amount of debt compared with other major real estate companies, which could weigh in its favor as it seeks support from Deutsche Bank and others. Representatives have been in talks with Rosemary Vrablic, a senior banker in the Deutsche Bank division that serves the ultrarich, about delaying or reducing its loan payments. Ms. Vrablic has been working with Mr. Trump for nearly a decade and, before that, worked closely with Jared Kushner, Mr. Trump’s son-in-law and senior adviser.

While Ms. Vrablic is in charge of the bank’s relationship with the company, changes to the terms of the loans could generate acrimony inside Deutsche Bank and are likely to be vetted by senior executives responsible for protecting the bank’s reputation, according to a person familiar with the deliberations.