WASHINGTON — A long-simmering feud within the White House broke into the open on Monday as a lawyer for John R. Bolton, President Trump’s former national security adviser, filed a motion trying to keep Mick Mulvaney, the president’s acting chief of staff, from joining a lawsuit over impeachment testimony.

Mr. Bolton’s lawyer argued in court papers that Mr. Mulvaney should not be allowed to jump into the existing lawsuit as a plaintiff because his interests are significantly different. But the legal schism underscored a broader rift between Mr. Mulvaney, who facilitated Mr. Trump’s effort to pressure Ukraine for damaging information about Democrats, and Mr. Bolton, who tried to resist it.

The lawsuit, filed by Charles M. Kupperman, a former deputy national security adviser and longtime associate of Mr. Bolton’s, asked a court to decide whether Mr. Kupperman should obey the president’s dictate or a House subpoena. While not technically a party to the lawsuit, Mr. Bolton, who left his post in September after clashing with Mr. Trump, is represented by the same lawyer, Charles J. Cooper, and is taking the same position as Mr. Kupperman in waiting for the court to decide whether he should testify or not.

Mr. Mulvaney’s effort to join the lawsuit late Friday night stunned many involved in the impeachment debate because he still works for the president. Rather than obey the president’s order not to cooperate with House investigators, as some other current administration officials have done, Mr. Mulvaney instead opted to ask a court to decide whether he should listen to his boss.

Mr. Mulvaney did not ask Mr. Bolton or Mr. Kupperman for permission to join the lawsuit nor did he give them a heads up. Mr. Bolton and his team considered it an outrageous move since they were on opposite sides of the Ukraine fight and did not want their lawsuit polluted with Mr. Mulvaney.

Not only did the motion filed Monday by Mr. Bolton’s camp seek to keep Mr. Mulvaney out of the lawsuit, it even advanced an argument that the acting chief of staff may have to testify before House impeachment investigators. The motion noted that in a briefing with reporters last month, Mr. Mulvaney appeared “to admit that there was a quid pro quo” before later trying to take back the admission, meaning that he might not have the right to defy a House subpoena since he had already discussed the matter in public.

“Accordingly, there is a serious question as to whether Mulvaney waived the absolute testimonial immunity claimed by the president,” the motion said.

The strongly worded motion reflected the divisions within Mr. Trump’s senior leadership team that played out largely behind closed doors for much of the year but have begun spilling out this fall during the House inquiry.

Even as some of the president’s advisers tried to help him extract help from Ukraine for his domestic political battles, others warned that it was an inappropriate abuse of his position. Mr. Mulvaney, occupying the southwest corner office in the West Wing, and Mr. Bolton, in the northwest corner office just down the hall, butted heads over the matter. At one point, according to testimony, 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.

Mr. Mulvaney’s court action on Friday led to the unusual spectacle of the president’s acting chief of staff seeking to join a lawsuit that names the president as one of the defendants. Mr. Mulvaney’s lawyers sought to finesse that by saying in their own legal papers that they wanted only to sue the House leaders.

Lawyers for the various parties are scheduled to hold a conference call later on Monday with United States Senior District Judge Richard J. Leon, who is overseeing Mr. Kupperman’s lawsuit.

The motion was meant to serve up the arguments in advance of the call and outlined four reasons Mr. Mulvaney should not be allowed to join the suit. One was Mr. Mulvaney’s public discussion of the case, as opposed to Mr. Kupperman, who has said nothing in public about it. Another is the fact that Mr. Mulvaney still works for the president while Mr. Kupperman does not.

The motion also argues that the factors are different for Mr. Kupperman, and by extension Mr. Bolton, who dealt exclusively with national security issues that are seen as more sensitive and therefore might have more weight justifying immunity from testimony, while Mr. Mulvaney does not.

And lastly, it noted that Mr. Kupperman does not take a position on who is right, the president or Congress, and “will remain neutral on the merits of the constitutional issue, while Mr. Mulvaney “has made it clear that he supports the executive” branch interpretation. “Mulvaney’s papers make very clear that he has aligned himself fully and completely with the president on the merits of the constitutional dispute before the court,” the motion said.

Michael S. Schmidt contributed reporting.