President Trump is in discussions with a veteran Washington lawyer who represented Bill Clinton during the impeachment process about joining the White House to help deal with the special counsel inquiry, according to four people familiar with the matter. The lawyer, Emmet T. Flood, met with Mr. Trump in the Oval Office this past week to discuss the possibility, according to the people. No final decision has been made, according to two of the people. Should Mr. Flood come on board, the two people said, his main duties would be a day-to-day role helping the president navigate his dealings with the Justice Department. Two people close to the president said that the overture to Mr. Flood did not indicate any new concerns about the inquiry. Still, it appears, at the least, to be an acknowledgment that the investigation is unlikely to end anytime soon. Mr. Flood would not replace Ty Cobb, the White House lawyer who since the summer has taken the lead role in dealing with the special counsel, Robert S. Mueller III. But Mr. Cobb has told friends for weeks that he views his position as temporary and does not expect to remain in the job for much longer. Mr. Cobb’s primary task — producing documents for Mr. Mueller and arranging for White House aides to meet with prosecutors — is largely complete. Mr. Trump’s personal lawyers have been handling negotiations with Mr. Mueller over the terms of a presidential interview. Mr. Flood had been on the wish list of some of the president’s advisers to join his legal team last year, and he is the only person the White House has been in contact with about such a leading role. White House officials did not immediately respond to a request for comment. Mr. Flood declined to comment. This is not the first time that the president’s advisers have considered a job for Mr. Flood, who worked in the White House counsel’s office under George W. Bush and represented Vice President Dick Cheney. 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. Other advisers have urged the president to make Marc E. Kasowitz, his longtime New York lawyer, with whom he had a falling out months ago, his lead lawyer again. Since Mr. Kasowitz has receded from the lead role, the president’s legal team has been composed of a crew of free agents, such as Mr. Cobb. Through intermediaries, Mr. Trump’s advisers have reached out to prominent lawyers to feel out their interest in joining his legal team. Most have expressed no interest.