But Mr. Trump has flouted his own lawyer, bragging to people around him that no one can control his actions.
Mr. Trump also privately criticized Mr. Kasowitz’s flat performance at a June 8 news conference after the Capitol Hill testimony of James B. Comey, the former F.B.I. director.
And angry, expletive-laced emails that Mr. Kasowitz sent to a critic were published this week by ProPublica, a nonprofit investigative news website.
But the main issue, a person close to Mr. Trump said, was proximity. Mr. Kasowitz commutes from his home in New York City and has been unable to give the president the seven-day-a-week commitment he requires as the investigation consumes a greater percentage of his time, aides said.
Emails sent to Mr. Kasowitz and his spokesman were not immediately returned.
For weeks, Donald F. McGahn II, the White House counsel, has been urging Mr. Trump to bolster his in-house team with a seasoned lawyer to help deal with an expected flood of document requests from the Justice Department’s special counsel, Robert S. Mueller III.
The idea was to have someone who could deal directly with Mr. Trump on the matter and coordinate with his personal lawyer, which would allow Mr. McGahn and the rest of the staff to tend to the day-to-day work of keeping the White House functioning, a White House official added.
In recent days, hiring such a player became more urgent, people close to the decision said, with the admission by the president’s eldest son, Donald Trump Jr., that he met during the heat of last year’s presidential campaign with a Russian lawyer who promised to provide damaging information about Hillary Clinton.
But many of Washington’s best criminal defense lawyers have appeared reluctant to work for a president known to ignore legal counsel and a family that has repeatedly failed to fully disclose contacts with people linked to Russia.
Mr. Trump and his team met Mr. Cobb this week and were impressed by his aggressiveness, according to a person who spoke with the president.
The president’s team also recently contacted another high-profile Washington lawyer, Emmet Flood, a partner at Williams & Connolly, who had previously rejected Mr. Trump’s entreaties.
Ms. Gorelick, a Democrat who began working with Mr. Kushner after advising Secretary of State Rex W. Tillerson during his confirmation hearings, denied reports that she was quitting Mr. Kushner’s team. She will hand over representation on matters related to the Russia investigation to another powerful Democratic lawyer, Abbe D. Lowell.
“As we have stated, once Bob Mueller and three of our partners left the firm to form the Special Counsel’s Office, we advised Jared to get independent legal advice on whether to continue with us as counsel,” Ms. Gorelick wrote in a statement on Friday.
“As a result of this process, Jared decided that Abbe would represent him in the Russia-related inquiries. We are currently helping Abbe’s team. We will continue to work on the matters for which we were originally retained, with regard to ethics compliance, the SF-86 process and related issues.”
Mr. Kushner, who had previously failed to reveal a meeting with Russia’s ambassador to the United States on a disclosure form, was briefly present during the meeting with the president’s son in which the anti-Clinton research was discussed.