Ms. Crowley fact-checked Mr. Romney when he wrongly claimed it took Mr. Obama 14 days to call an attack in Benghazi, Libya, an “act of terror.”

The meeting between Mr. Parscale and Mr. Fahrenkopf ended after 45 minutes with no resolution.

Since then, Mr. Parscale has told people that he was investigating other options for hosting the debates. It is not clear which outside firms he or other officials are talking to, and the campaign declined to provide any details.

“We want to have debates that are fair and are more geared toward informing the American people than to boosting the careers of the moderators,” Tim Murtaugh, a spokesman for Mr. Trump’s campaign, said of the meeting.

The commission has scheduled three presidential debates, to be held on college campuses in late September and October, as well as one vice-presidential debate.

Mr. Trump has been discussing the possibility of sitting out the general election debates for months. He has harbored bad feelings about the debate commission since the 2016 election, when he accused them of putting him at a disadvantage “on purpose” by giving him a “defective mic” at the first debate. (Mr. Trump was clearly audible to television viewers, but the commission said a technical malfunction affected the volume of his voice in the debate hall.)

After The New York Times reported that Mr. Trump had discussed the possibility of sitting the debates out, he wrote on Twitter that he wanted to face off against his eventual Democratic opponent. But he said that “the problem is that the so-called Commission on Presidential Debates is stacked with Trump Haters & Never Trumpers.”

He added that “there are many options, including doing them directly & avoiding the nasty politics of this very biased Commission. I will make a decision at an appropriate time but in the meantime, the Commission on Presidential Debates is NOT authorized to speak for me (or R’s)!”