WASHINGTON (AP) — Energy Secretary Rick Perry said Wednesday he had no interest in becoming the next head of the Department of Veterans Affairs, flatly rejecting speculation that he would soon take over the position amid rapidly eroding White House support for embattled VA Secretary David Shulkin.

Two administration officials told The Associated Press that Shulkin's position is growing more precarious and that he could be out of a job within the week, but cautioned that nothing was finalized. The officials spoke on condition of anonymity to discuss a sensitive personnel matter.

Shulkin, a form...