While the White House publicly backed Mr. Holder as he tried to smooth over the latest uproar amid new speculation about his future, some in the West Wing privately tell associates they wish he would step down, viewing him as politically maladroit. But the latest attacks may stiffen the administration’s resistance in the near term to a change for fear of emboldening critics.

The White House views the attacks on Mr. Holder as a “political agenda” and “would not hasten the departure of someone who’s competent and runs the department and is a friend because there’s a drumbeat,” said William M. Daley, a former White House chief of staff under Mr. Obama. “Whoever Barack Obama puts in there, these people will try to drumbeat him out of there, no matter what.”

But that does not mitigate the frustration of some presidential aides. “The White House is apoplectic about him, and has been for a long time,” said a Democratic former government official who did not want to be identified while talking about friends.

Some advisers to Mr. Obama believe that Mr. Holder does not manage or foresee problems, the former official said. “How hard would it be to anticipate that The A.P. would be unhappy?” the former official said. “And then they haven’t defended their position.”