WASHINGTON—The House Judiciary Committee is preparing to file a lawsuit in the coming days to enforce a subpoena for testimony by former White House counsel Don McGahn, as the panel has so far failed to reach a deal with the White House over a central aspect of his testimony: whether it would be public or private.

In conversations with Deputy White House Counsels Patrick Philbin and Michael Purpura, lawyers for the House Judiciary Committee said the panel would limit the scope of its questioning to what was covered in former...