"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."
The interaction followed President Donald Trump's abrupt exit from a meeting with Democratic congressional leaders.
Pelosi earlier in the day accused the president of engaging in a "cover-up" by blocking White House aides from providing testimony and responding to document requests from congressional investigations.
In a last-minute press event at the Rose Garden, Trump said he "walked into the room and I told Sen. Schumer and Speaker Pelosi, 'I want to do infrastructure' ... but we can't do it under these circumstances."
Democratic leaders said they were ready to make a deal on a $2 trillion plan to rebuild the country's infrastructure, including roads, bridges and airports.