Other companies may be able to tap into a $500 billion rescue fund that will be administered by the Treasury Department. But the economic bailout package, which Mr. Trump signed into law last week, specifically barred the president and his family from access to that money.

Late last month, Mr. Trump’s representatives contacted their relationship managers in Deutsche Bank’s New York private-banking division, which caters to wealthy customers. They wanted to discuss the possibility of delaying payments on some of the hundreds of millions of dollars of outstanding loans that the Trump Organization has from the bank, according to a person briefed on the talks. The discussions are continuing.

Around the same time, Ed Raymundo, a Trump Organization executive in Florida, emailed and called Palm Beach County officials to discuss whether they planned to keep collecting monthly rent payments on land that Mr. Trump’s company leases from the county, according to people familiar with the talks.

The Trump Organization has a small amount of debt compared with other major real estate companies, which could weigh in the company’s favor as it seeks accommodations.

Deutsche Bank has lent Mr. Trump and his companies about $2 billion since 1998, the only mainstream financial institution consistently willing to do business with Mr. Trump and his companies. At the time he became president, Mr. Trump owed the bank about $350 million, including on loans to buy and renovate the Doral golf resort near Miami and to develop a luxury hotel in the Old Post Office building in Washington.

Both properties are suffering in the economic shutdown. In response to Miami-Dade County’s rules, the Doral resort has ceased all operations, while the Washington hotel continues to operate, albeit with few guests and with its restaurant and bar closed. The Trump Organization rents the Washington property from the federal government, and the company had been soliciting bids from potential buyers for the lease, a process that is now on hold, The Washington Post reported.