The younger Mr. Trump’s allies quickly declared victory on Tuesday, noting that the deal was essentially what he had asked for at the outset when discussions over a return interview with the committee began months ago.

“This compromise shows all Don wanted was for the committee to be reasonable,” said Cliff Sims, a former White House communications aide and an ally of Mr. Trump’s. “Demanding unlimited time and scope was absurd — the guy’s already testified before Senate Intel for nine hours, after all.”

A person close to the younger Mr. Trump, who in 2018 was a highly in-demand Republican surrogate on the campaign trail, said he was grateful for the support that some Republican senators and members of the House gave him against Mr. Burr, and that he would remember it when the 2020 campaigns begin.

But the agreement also provides Mr. Burr with an off-ramp from the confrontation. If the younger Mr. Trump had refused to appear, the chairman would have faced a painful choice between initiating contempt of Congress proceedings against the president’s eldest son and undercutting the independence of his two-year investigation of Russian election interference by letting him defy him.

President Trump personally questioned why his son would be subpoenaed after the special counsel, Robert S. Mueller III, had been unable to prove a conspiracy between his campaign and Russia. Republican allies, including Senator Lindsey Graham of South Carolina, chairman of the Judiciary Committee, were on board. On Monday, Mr. Graham said if he was the younger Mr. Trump’s lawyer, he would advise him not to talk.

Asked about the subpoena on Monday, Mr. Trump said he viewed it as unfair.

“It’s really a tough situation because my son spent, I guess, over 20 hours testifying about something that Mueller said was 100 percent O.K.,” he told reporters at the White House. “And now they want him to testify again. I don’t know why. I have no idea why, but it seems very unfair to me.”