U.S. Speaker of the House Rep. Nancy Pelosi (D-CA) speaks to members of the media after a House Democrats meeting at the Capitol May 22, 2019 in Washington, DC.

House Speaker Nancy Pelosi snapped at White House counselor Kellyanne Conway after a tense, aborted meeting on infrastructure Wednesday.

"I don't talk to staff. I talk directly to the president," Pelosi said after Conway asked her a question, according to a senior administration official who was in the room.

Later, a Pelosi staffer told CNBC that the speaker had said something different. "I'm responding to the president, not staff," Pelosi said, according to her deputy chief of staff, Drew Hammill.

In a Fox News interview Thursday morning, Conway gave her own account of the tiff.

"I said, respectfully, 'Madame Speaker, would you like to address some of the specifics the president talked about?'" Conway said. "And [Pelosi] said, 'I don't – I talk to the president, I don't talk to staff.'"

Conway continued: "Because let's face it, she's the sixth most rich member of Congress, she treats everybody like [they're] her staff. She treats me like I'm either her maid or her driver or her pilot or her makeup artist, and I'm not. And I said to her, 'How very pro-woman of you,' per usual. Because she's not very pro-woman, she's pro-some women, a few women."