WASHINGTON—Within a day of the release of the Mueller report last month, President Trump sought to have former White House counsel Don McGahn declare he didn’t consider the president’s 2017 directive that he seek Robert Mueller’s dismissal to be obstruction of justice, but Mr. McGahn rebuffed the request, according to people familiar with the matter.

Mr. Trump has publicly denied asking Mr. McGahn to fire the Russia probe special counsel since the release of the report. Mr. Mueller’s report detailed that directive, and a subsequent request by Mr. Trump that Mr. McGahn deny that conversation ever happened, and said that Mr. McGahn rebuffed both. Last month, Mr. Trump tweeted: “If I wanted to fire Mueller, I didn’t need McGahn to do it, I could have done it myself.”

Privately, Mr. Trump asked White House special counsel Emmet Flood to inquire whether Mr. McGahn would release a statement asserting that he didn’t believe those interactions with the president—and Mr. Trump’s subsequent efforts to have Mr. McGahn deny news reports about that request—amounted to obstruction, the people familiar with the matter said. Mr. Flood didn’t respond to a request for comment.

William Burck, a lawyer for Mr. McGahn, said in a statement about the request: “We did not perceive it as any kind of threat or something sinister. It was a request, professionally and cordially made.”

Mr. McGahn turned down the request because he didn’t want to weigh in on the totality of evidence in the report beyond his own testimony, and didn’t want to comment on his own testimony in isolation, the people said. Mr. McGahn also didn’t view his personal opinion as relevant, because Attorney General William Barr had already said he didn’t believe the evidence in Mr. Mueller’s report amounted to obstruction of justice, the people said.

Mr. Flood, as he sought the statement from Mr. McGahn, pointed to previous assertions by Mr. Burck that if Mr. McGahn believed Mr. Trump had committed a crime, he would have resigned his post, the people said.

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Last August, following reports that Mr. McGahn had spent 30 hours talking to Mr. Mueller’s investigators, Trump lawyer Jay Sekulow reached out to Mr. Burck to inquire about the scope and nature of Mr. McGahn’s testimony to Mr. Mueller, according to people familiar with the conversation. John Dowd, who headed the president’s legal team, had also contacted Mr. Burck previously to inquire about Mr. McGahn’s testimony and whether he had told Mr. Mueller that Mr. Trump committed any crimes, the people said.

Mr. McGahn told investigators that he prepared to resign in June 2017 rather than follow Mr. Trump’s directive to call the deputy attorney general and ask him to dismiss Mr. Mueller, but stayed in his post when Mr. Trump didn’t press him on the matter. Mr. McGahn left his White House position in October 2018.

Mr. Trump’s effort to have Mr. McGahn make a public statement after the report’s release could intensify efforts by Democratic lawmakers to compel him to testify about his interactions with Mr. Trump. The White House on Tuesday directed Mr. McGahn to rebuff a subpoena from the House Judiciary Committee for documents related to Mr. Mueller’s investigation, saying the requested documents implicated executive privilege.

Mr. Trump has said he will fight all congressional subpoenas to his current and former aides, telling reporters last month: “It’s enough.”

The House Judiciary Committee voted to hold Attorney General William Barr in contempt of Congress after he defied their request for an unredacted Mueller report. WSJ reporters break down what legal options Congress has going forward to enforce a subpoena. Photo Illustration: Nailah Morgan

House Judiciary Committee Chairman Jerrold Nadler (D., N.Y.), in a letter to Mr. Burck on Tuesday evening, said he expected the committee would vote to hold Mr. McGahn in contempt if he didn’t appear before the panel on May 21. If that were to happen, the matter would be likely be litigated in the courts.

Mr. Trump and his allies have criticized Mr. McGahn since the report’s release. Trump lawyer Rudy Giuliani called Mr. McGahn “hopelessly confused” in a CNN interview. Mr. Burck has defended the episodes in the report involving his client as “accurately described” and called Mr. Giuliani’s comments a “mystery.”

A statement from Mr. McGahn declaring that he didn’t believe Mr. Trump committed a crime could have bolstered the White House’s broader public argument for Mr. Trump’s innocence, particularly as Congress continues to investigate Mr. Trump for obstruction and other matters related to the probe of Russian interference in the 2016 presidential election and Moscow’s ties to the Trump campaign. Mr. Trump has publicly declared the Mueller report exonerates him and has denied obstructing justice.

Mr. Mueller in his report said he wasn’t exonerating Mr. Trump and wrote: “The evidence we obtained about the President’s actions and intent presents difficult issues that prevent us from conclusively determining that no criminal conduct occurred.” He said he decided not to make a decision on whether Mr. Trump obstructed justice in part because of a Justice Department policy that a sitting president cannot be indicted.

According to the Mueller report, Mr. Trump called Mr. McGahn at home in June 2017—a month after Deputy Attorney General Rod Rosenstein had appointed Mr. Mueller—and directed him to call Mr. Rosenstein and say Mr. Mueller had conflicts that precluded him from serving as special counsel. “You gotta do this. You gotta call Rod,” the president said.

Mr. McGahn was “perturbed” by Mr. Trump’s order, according to the report, and prepared to resign rather than carry it out, telling then-chief of staff Reince Priebus the president had asked Mr. McGahn to “do crazy shit.” He told Mr. Mueller’s investigators that he and other White House aides saw the alleged conflicts—which included Mr. Mueller’s effort to obtain a refund from one of Mr. Trump’s golf clubs in 2011—as “silly” and “not real,” and that they had said as much to Mr. Trump, according to the report.

After the New York Times reported in January 2018 that Mr. Trump had directed Mr. McGahn to have the special counsel dismissed, Mr. Trump publicly denied it and sought to have aides, including his personal lawyer, his press secretary and his staff secretary, ask Mr. McGahn to dispute the reports, at one point threatening to fire Mr. McGahn, according to the Mueller report. The next month, he met face-to-face with Mr. McGahn and asked him: “Did I say the word fire?” Mr. McGahn told Mr. Trump that he understood the conversation as “Mueller has to go.”

—Brody Mullins contributed to this article.

Write to Rebecca Ballhaus at Rebecca.Ballhaus@wsj.com