WASHINGTON—The House Oversight Committee chairman asked Manhattan federal prosecutors whether a Justice Department policy barring charges against a sitting president played any role in their decision not to indict President Trump for campaign-finance violations in which he was implicated.

In a letter Friday to Audrey Strauss, the deputy U.S. Attorney in Manhattan, Rep. Elijah Cummings (D., Md.) asked whether prosecutors had identified evidence of criminal conduct by Mr. Trump and whether the Justice Department policy had “played...