WASHINGTON—A former West Wing aide tape recorded conversations with President Trump, creating a fresh issue for a White House already displeased over the level of leaks of internal deliberations, according to two people familiar with the matter.

Omarosa Manigault-Newman, a former contestant in Mr. Trump’s reality TV program, taped high-level administration officials and Mr. Trump during her tenure as an assistant to the president, these people said.

Ms....