Perhaps more consequentially for Mr. Shulkin, Mr. Trump’s advisers say White House officials began to believe that Mr. Shulkin misled them about what would be in the report, which ushered in a new round of negative news media reports on the president’s cabinet. He made matters worse, they said, when he botched attempts to clean up the story and went public with fears about appointees “trying to undermine the department from within.”

As stories about potential replacements began to circulate in mid-March, Mr. Trump summoned Mr. Shulkin to the Oval Office for what one person briefed on the discussion described as a tense meeting. An aide to Mr. Shulkin disputed that characterization.

Mr. Shulkin has played down speculation about his departure as he has testified about the department’s budget request for the coming fiscal year and met privately with lawmakers to discuss pending legislation to remake the department’s approach to private care. He was scheduled to host a regular meeting on Thursday with leaders of the nation’s largest veterans groups.

But he has proceeded with a curtailed public profile. The secretary was not on hand for a veterans event at the White House on Friday, for instance, and he is now hurried in and out of events by a security detail without taking questions from the press. Those who have spoken with the secretary said he had made it clear that he wanted to continue in the department’s top job, but was increasingly isolated.

Ashleigh F. Barry, a spokeswoman for the secretary, said on Tuesday only that Mr. Shulkin was focused on the work of the department.

The speculation and mixed messages coming out of the White House have flummoxed key lawmakers in both parties, who continue to support the secretary, as well as many of the major veterans groups, which are congressionally chartered and largely apolitical. Organizations such as the Veterans of Foreign Wars and American Legion hold considerable sway over veterans policy in Washington, and they solidly back Mr. Shulkin.

Though the groups were generally displeased with the findings turned up by the inspector general, they see the secretary as a pragmatic and effective leader who understands the challenges facing the department’s sprawling health care system and has taken concrete steps to address them.