Mr. Trump’s lawyers have homed in on his account of Mr. Trump’s ordering the special counsel fired in June 2017 and later attempts to get Mr. McGahn to recant what he told investigators about the episode. The encounters — lawmakers are pursuing documents about them — are two of the most damning examples of potential obstruction of justice laid out in the Mueller report and show the clearest example of Mr. Trump attempting to influence witness testimony.

Days after a New York Times article in January 2018 revealed that the president directed Mr. McGahn to have the Justice Department fire Mr. Mueller, Mr. Trump told a senior aide he was considering firing Mr. McGahn. The president also wanted Mr. McGahn to create a paper trail denying his own account by writing a letter to keep on file, “something beyond a press statement to demonstrate that the reporting was inaccurate,” according to the report.

Mr. McGahn refused, setting up an Oval Office confrontation with the president. Mr. Trump denied that he ordered Mr. Mueller fired, saying he had instead wanted Mr. McGahn to alert the Justice Department to what the president contended were several conflicts of interest of Mr. Mueller, according to the report. But Mr. McGahn reminded the president that he had indeed told him to fire the special counsel, the report said. The president again asked Mr. McGahn to issue a correction to the Times article, but Mr. McGahn refused.

“McGahn thought the president was testing his mettle to see how committed McGahn was to what happened,” Mr. Mueller’s investigators wrote.

Mr. Giuliani has picked up where Mr. Trump left off, insisting that the president never told Mr. McGahn to have Mr. Mueller fired. He said the president was only venting.

Mr. Giuliani also falsely asserted that Mr. McGahn provided conflicting accounts to investigators.

“This is much more a sleight of hand by special counsel than by McGahn because it sounds like McGahn gave two to three different versions,” Mr. Giuliani said, adding that he thought Mr. Mueller’s report was a “disgrace” because it left out that Mr. Trump brought up the special counsel’s perceived conflicts of interest with other people without demanding he be fired.

Mr. Burck said his client was simply telling the truth under threat of perjury, not accusing the president of illegally obstructing justice.