The disparity highlights the fundamental tension between the natural impulse to protect a president’s health and the desire by most politicians to project that they are not receiving any special treatment, Robert Dallek, the presidential historian, said in an interview Friday.

By flaunting his own access to tests and making false claims about the availability of testing, Mr. Trump, he said, was only hurting his own credibility with voters.

“When you add it to the fact that people on Capitol Hill, who after all form an essential part of the government as well, cannot get testing as readily, it just underscores the feeling that this man is principally self-serving,” Mr. Dallek said. “It is not a good impression for the White House to convey.”

The White House appears to agree. Shortly after 11 p.m. Friday, Mr. Trump’s health secretary, Alex M. Azar II, made an announcement on Twitter: “Good news: as the Senate reconvenes to do important work for the American people during this public health crisis, we have now received an initial request and are sending 3 Abbott point of care testing machines and 1,000 tests for their use.”