WASHINGTON—The House Judiciary Committee believes it has evidence that President Trump asked Matthew Whitaker, at the time the acting attorney general, whether Manhattan U.S. attorney Geoffrey Berman could regain control of his office’s investigation into Mr. Trump’s former lawyer and his real-estate business, according to people familiar with the matter.

Mr. Berman, a former law partner of Trump attorney Rudy Giuliani who in 2016 donated to the Trump campaign and whom the president personally interviewed for the U.S. attorney...