Another person familiar with the latest moves said that Mr. Paoletta was considered, but that West Wing officials decided they were pleased with the hiring of Pam Bondi, a former attorney general of Florida, and Tony Sayegh, a Republican strategist, both of whom began full time this week.

The latest personnel struggle echoed an attempt by Mr. Mulvaney several weeks ago to hire former Representative Trey Gowdy, a fellow South Carolina Republican, to join the president’s legal team. Mr. Cipollone and others were said to take issue with the idea, concerned it was an effort by Mr. Mulvaney to run his own legal team. Mr. Cipollone told allies he had no such concerns, but eventually, Mr. Gowdy bowed out, facing an issue with a ban on former House members lobbying Congress.

Despite his own tenuous job status, Mr. Mulvaney has privately told associates in recent days that there is no easy way for Mr. Trump to fire him in the midst of the impeachment fight, the implication being that he knows too much about the president’s pressure campaign to force Ukraine to provide incriminating information about Democrats.

The court fight between Mr. Mulvaney and Mr. Bolton on Monday brought their long-running feud into the open. Mr. Mulvaney was among those facilitating the Ukraine effort, while Mr. Bolton was among those objecting to it. At one point, according to testimony in the impeachment inquiry, Mr. Bolton declared that he wanted no part of the “drug deal” Mr. Mulvaney was cooking up, as the then national security adviser characterized the pressure campaign.

Their clash was just one of many inside Mr. Trump’s circle spilling out into public in recent days. The legal conflict on Monday came just a day before Nikki R. Haley, the president’s former ambassador to the United Nations, plans to publish a memoir accusing Mr. Trump’s former secretary of state, Rex W. Tillerson, and former chief of staff, John F. Kelly, of conspiring behind his back while in office. Her account in effect is a mirror image of another book coming out this month by an anonymous senior administration official describing how concerned aides mounted their own internal resistance to Mr. Trump.

Mr. Kelly disputed Ms. Haley in a statement on Sunday and Mr. Tillerson added his own refutation on Monday. “During my service to our country as the secretary of state, at no time did I, nor to my direct knowledge did anyone else serving along with me, take any actions to undermine the president,” Mr. Tillerson said in a statement.

While he offered Mr. Trump frank advice, he said, once the president made a decision, he did his best to carry it out. “Ambassador Haley was rarely a participant in my many meetings and is not in a position to know what I may or may not have said to the president,” Mr. Tillerson added.