WASHINGTON—Robert Mueller will testify in open session next month before a pair of House committees, in a victory for the Democratic leaders who had long sought his testimony in public.

House Judiciary Committee Chairman Jerrold Nadler (D., N.Y.) and House Intelligence Committee Chairman Adam Schiff (D., Calif.) said that the special counsel had agreed to testify in open session on July 17 pursuant to a subpoena.

The...