Energy Secretary Rick Perry says he has no interest in leading the Department of Veterans Affairs.
The VA secretary also addressed reports last week claiming Shulkin placed an armed guard outside his office door.
The White House is declining to offer an endorsement of embattled Veterans Affairs Secretary David Shulkin, who is under fire for ethics violations.
"I've publicly acknowledged that the distraction that has happened that you've talked about is something that I deeply regret", Shulkin told a House Appropriations Committee subpanel.
Kelly, who found out about The Times's article after he was contacted by reporters, reportedly called Shulkin following the interview and believed the secretary exploited the White House meeting, Axios reported. Perry arrived first and told CNN's Rene Marsh that he isn't interested in the VA job. "There is the "people who have caused him embarrassing headlines" bucket, which is Ben Carson, the HUD secretary, and the VA secretary David Shulkin, with reports that they have been, you know, using taxpayer money lavishly".
Shulkin is under criticism for ethics violations related to a trip he took to Europe with his wife.More news: Man United captain Carrick to retire
If Dr. Shulkin is removed, it would be the latest in a series of prominent departures from the Trump administration.
Two administration officials told The Associated Press on Wednesday that Shulkin's position is precarious and that he could be out of a job within the week, but they cautioned that nothing was finalized.
The official said the discussions of replacing the VA head are "not very far along" and a different person familiar with the matter said other candidates are in the mix.
"Every Cabinet member has a security detail that is armed", he said.
The uncertainty over Shulkin's status has left the VA, which employs 360,000 people, virtually paralyzed, with members of Congress confused over who's in charge and what's being done to improve care for 9 million military veterans in more than 1,700 government-run health facilities. Just two lawmakers, Republican Reps.
But behind the scenes for months, Shulkin has been at odds with members of the department, installed by the Trump administration, a dispute over how much veterans should be able to seek private care outside the VA health care system. The IG found that Shulkin's chief of staff doctored emails to justify his wife accompanying him at taxpayer expense.