It is not clear whether the president supports the move. The Department of Veterans Affairs referred questions about its leadership, the tensions in the department and Mr. Leinenkugel’s email to the White House, which declined to comment.

The fight over the leadership of the department is part of a long-running battle over how to deliver health care to the nation’s veterans. The department currently operates its own health system, with more than 1,200 hospitals and clinics across the country where about nine million veterans receive treatment at little or no cost to them. Some conservatives, including some advisers to the White House, favor gradually dismantling that system and allowing veterans to choose to receive taxpayer-subsidized care from private doctors instead.

Veterans’ groups have overwhelmingly opposed that idea. But Mr. Trump promised during his election campaign that “vets will have the right to go to a V.A. facility or the right to see a private doctor or clinic of their choice — whatever is fastest or best for the vet.”

Policymakers in both parties have said that offering that kind of unrestricted choice between the public system and private providers would be too expensive. They have generally favored a more measured approach that would allow the department to approve the use of private care when waiting times are too long at veterans’ hospitals. Dr. Shulkin has also backed this plan.

Dr. Shulkin said in the interview that he was determined to remain in his post. Though one lawmaker called for his resignation in the wake of the travel report, others were cordial to Dr. Shulkin at a budget hearing Thursday morning and did not bring up the report.

He said he was investigating a number of political appointees in his department for misconduct and possible removal. On Thursday, he spoke directly to the White House chief of staff, John F. Kelly, about concerns that political appointees were trying to undermine his agency, according to department officials.

“If there are people here who don’t want the V.A. to succeed, I want them out,” Dr. Shulkin said in the interview.