His visits to the White House have raised questions about the blurry line between public and private interests for a president facing legal issues. Mr. Kasowitz in recent days has advised White House aides to discuss the inquiry into Russia’s interference in last year’s election as little as possible, two people involved said. He told aides gathered in one meeting who had asked whether it was time to hire private lawyers that it was not yet necessary, according to another person with direct knowledge.

Such conversations between a private lawyer for the president and the government employees who work for his client are highly unusual, according to veterans of previous White Houses. Mr. Kasowitz bypassed the White House Counsel’s Office in having these discussions, according to one person familiar with the talks, who like others requested anonymity to discuss internal matters. And concerns about Mr. Kasowitz’s role led at least two prominent Washington lawyers to turn down offers to join the White House staff.

“The president’s private lawyer is representing only his interests, not the interests of the United States government or the individual interests of the White House staff,” said Robert F. Bauer, who was White House counsel under President Barack Obama.