WASHINGTON—The House Judiciary Committee reached an agreement to gain access to evidence special counsel Robert Mueller collected on whether the president acted improperly, the panel’s chairman said, and backed off a plan to hold Attorney General William Barr in contempt of Congress for blocking the release.

The Justice Department is “providing us with key evidence that the special counsel used to assess whether the president and others obstructed justice or were engaged in other misconduct,” said House Judiciary Committee...