Senior administration officials say Shulkin is increasingly on thin ice with the president after a bruising internal report found ethics violations in connection with the secretary's trip to Europe with his wife last summer.
Shulkin, who has been telling the media that Trump political appointees in the Department of Veterans Affairs are trying to undermine him, was reportedly told by Kelly to stop fighting with staff and talking with the media. Shulkin has since reimbursed the department for his wife's airfare.
Trump put host Pete Hegseth on speakerphone so he could tell Shulkin his take on new laws proposed in Congress that would expand access to private healthcare for veterans, two sources with knowledge of the conversation told Axios.
An Energy Department spokeswoman, asked about reports Perry is being considered to replace Shulkin, said: "I have no comment at this time".
Trump and Perry discussed the possible move during a lunch at the White House Monday.
Possible replacements for Energy Secretary show Trump also is unfamiliar with the responsibilities of the Energy Secretary.
Hegseth, a veteran of the wars in Afghanistan and Iraq, competed against Shulkin for the job of heading Veterans Affairs. He was the only holdover from the previous administration after Trump tapped him as the new VA secretary. Trump said during a press conference early last week that he likes "conflict" and having "having two people with different points of view" go head-to-head in front of him, adding that it helps him make decisions. Shulkin was the only one among Trump appointees to be unanimously confirmed by the Senate. Carson has since rescinded the order and promised to give a "full disclosure" of the supposed purchase.
Shulkin has expressed his regret for the distractions that have prevented the VA from focusing on serving veterans.
Trump has talked about firing Shulkin but was still weighing the question Tuesday, the people said.