WASHINGTON — An exasperated President Trump picked up the phone to call the White House counsel, Donald F. McGahn II, last Sunday. Tell the F.B.I. they can investigate anything, he told Mr. McGahn, because we need the critics to stop.

Not so fast, Mr. McGahn said.

Mr. McGahn, according to people familiar with the conversation, told the president that even though the White House was facing a storm of condemnation for limiting the F.B.I. background check into sexual misconduct allegations against Judge Brett M. Kavanaugh, a wide-ranging inquiry like some Democrats were demanding — and Mr. Trump was suggesting — would be potentially disastrous for Judge Kavanaugh’s chances of confirmation to the Supreme Court.

It would also go far beyond the F.B.I.’s usual “supplemental background investigation,” which is, by definition, narrow in scope.

The White House could not legally order the F.B.I. to rummage indiscriminately through someone’s life, Mr. McGahn told the president. And without a criminal investigation to pursue, agents could not use search warrants and subpoenas to try to get at the truth.