A White House spokesman did not respond to an email seeking comment about whether they will instruct Ms. Hicks to follow Mr. McGahn’s lead, or what legal grounds they might invoke to do so. But for Ms. Hicks, the options are fraught.

Witnesses have generally followed the White House lead, in part because of institutional concerns about areas that could be viewed as covered by executive privilege. But if Ms. Hicks does not cooperate, she would potentially be in legal jeopardy with the House.

The likeliest possibility would be a compromise, where she would submit to an interview as long as certain topics are off limits. More recently, Mr. Trump’s eldest son, Donald Trump Jr., reached a deal with the Senate Intelligence Committee to come in for a limited interview, after he balked at a subpoena.

Ms. Hicks declined to comment, as did her lawyer.

While serving in the White House, Ms. Hicks never appeared on camera, unless it was at Mr. Trump’s side, and never gave an on-camera interview. Since leaving Washington, she has maintained an equally low profile, working at a spinoff company from Fox News. But the Mueller report provided an in-depth, and rare, look at Ms. Hicks’s interactions with the president.

Ms. Hicks was mentioned on 28 pages in the report. Three of those are related to possible conspiracy between Russian officials and the Trump campaign, and the rest to the obstruction investigation. They paint a picture of an adviser who was more of a witness to the president’s frustrations with the investigations into his campaign and his own conduct, rather than someone who was an active participant in any discussions of what to do about them.

Ms. Hicks comes across in her interviews with the F.B.I. as trying to alert Mr. Trump to the possible news media reaction he might face to any new information about what took place in the campaign.

For instance, she is described telling Mr. Trump that emails existed related to Mr. Kushner and a meeting with a Kremlin-linked lawyer that took place in June 2016 at Trump Tower. The report describes Ms. Hicks looking at the emails at Mr. Kushner’s lawyer’s office, and being “shocked by the emails because they looked ‘really bad.’”