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(CNN) —  

During a tense exchange late Wednesday, Sen. Kamala Harris pressed Supreme Court nominee Brett Kavanaugh about whether he had discussed special counsel Robert Mueller’s investigation with anyone, including any individuals at the law firm founded by the President’s personal attorney.

On the second day of confirmation hearings, the California Democrat asked the appellate judge whether he had “ever discussed special counsel Mueller or his investigation with anyone.”

After answering “with other judges I know,” Kavanaugh was asked if he had discussed the probe with anyone who works at Kasowitz Benson Torres, the New York law firm founded by President Donald Trump’s personal attorney Marc Kasowitz.

Kavanaugh replied that he’s unsure he knows everyone who works at that law firm and asked the senator if there was a specific person she was talking about.

“I’m not remembering, but I’m happy to be refreshed or if you want to tell me who you’re thinking of,” Kavanaugh said.

“You’ve been speaking for almost eight hours to this committee about all sorts of things you remember,” Harris said. “How can you not remember whether or not you had a conversation about Robert Mueller or his investigation with anyone at that law firm?”

Later, Kavanaugh said he would like to know the specific person Harris was thinking of.

Harris fired back, “I think you are thinking of someone, and you don’t want to tell us.”

Republican Sen. Mike Lee of Utah then interjected, raising a point of order.

“If there is a list of names he can be given of the lawyers to whom she is referring, I think that’s fine,” Lee said. “But I think it’s unfair to suggest that an entire law firm should be imputed into the witness’s memory when he doesn’t know who works at the law firm.”

Harris, who has been mentioned as a 2020 presidential contender, returned to questioning Kavanaugh, asking him again if he had a conversation with anyone about Mueller’s investigation.

“The fact that it’s ongoing, it’s a topic in the news every day, I talk to fellow judges about it. It’s the courthouse in the District of Columbia. So, I guess, the answer to that is yes,” the Supreme Court nominee said.

Once again, Harris asked Kavanaugh if he spoke with anyone employed at Kasowitz’s law firm about the special counsel investigation, arguing that he can answer the question without a list of its employees.

“I’d be surprised but I don’t know everyone that works at that law firm,” Kavanaugh said, adding, “I’m not remembering anything like that but I want to know a roster of people and I want to know more.”

Harris then moved on to other points of questioning, but not before she told Kavanaugh “clearly you’re not going to answer the question.”

Kavanaugh responded to the questioning again on Thursday saying he hasn’t had any “inappropriate” conversations about the special counsel’s Russia investigation “with anyone,” adding he doesn’t “recall any conversations of that kind with anyone.”

“I don’t recall any conversations of that kind with anyone at that law firm,” Kavanaugh said during the third day of hearings. “I haven’t had any inappropriate conversations about that investigation with anyone.”

The law firm also issued a statement to CNN on Thursday, saying, “There have been no discussions regarding Robert Mueller’s investigation between Judge Kavanaugh and anyone at our firm.”

Harris pushed back against the law firm’s denial, telling CNN, “They’re not under oath.”

Asked if she didn’t believe them, Harris said again, “They’re not under oath. The question was asked under oath.”

A Democratic aide told CNN that “we have reason to believe that a conversation happened and are continuing to pursue it.” No further details were provided by Harris to Kavanaugh during the hearing, and she said she would have questions for the record for Kavanaugh about the issue.

Currently a judge on the US Court of Appeals for the DC Circuit, Kavanaugh, 53, was nominated to the Supreme Court by Trump in July to replaced retired Justice Anthony Kennedy. If Kavanaugh is confirmed, he could establish a conservative majority on the nation’s highest court for decades. The third day of his confirmation hearings continues Thursday.

This story has been updated to reflect additional developments.

CNN’s Phil Mattingly, Daniella Diaz, Manu Raju and Dana Bash contributed to this report.