As recently as the summer, Mr. Flood, who currently works at the law firm Williams & Connolly, turned down an opportunity to represent Mr. Trump. It is not clear what has changed since then.

People close to Mr. Trump have long praised Mr. Cobb as having a deft touch with an often mercurial president. Throughout last year, Mr. Cobb kept Mr. Trump from publicly airing grievances against Mr. Mueller in part by telling him that the investigation would be wrapped up by December, or soon after — an assessment that proved too optimistic.

But there have been signs in recent months that Mr. Trump may be looking to shake up his legal team and change his approach to Mr. Mueller’s investigation. The president has polled his advisers and friends, asking them what they think of Mr. Cobb, who persuaded Mr. Trump to take a cooperative approach to the inquiry.

In private conversations, Mr. Trump has seesawed between expressing confidence in Mr. Cobb’s claim that the inquiry will wrap up in relatively short order and that he will be exonerated, and sounding frustrated with his team’s legal strategy.

Mr. Cobb has clashed with the White House counsel, Donald F. McGahn II, who believed that Mr. Cobb was too willing to hand over documents to Mr. Mueller when the White House could have shielded them by citing executive or attorney-client privilege. Officials familiar with the discussions with Mr. Flood said they were unrelated to tensions between Mr. McGahn and Mr. Cobb.

It is not clear what a shake-up would mean for John Dowd and Jay Sekulow, the two lawyers outside the White House who have also represented Mr. Trump since the summer. Mr. Dowd has been at the center of a string of embarrassing incidents, including one in which he wrote a tweet for Mr. Trump that raised new questions about whether the president had tried to obstruct the investigation into his former national security adviser, Michael T. Flynn.

One person close to the president who has urged him to dismiss Mr. Cobb and Mr. Dowd has been the Fox News personality Jeanine Pirro. Ms. Pirro, a personal friend of Mr. Trump’s and a former Westchester County district attorney, told Mr. Trump in an Oval Office visit months ago that Mr. Cobb and Mr. Dowd were leading him down a path toward his demise.