Washington (CNN) Senate Judiciary Chairman Lindsey Graham sent a letter to special counsel Robert Mueller on Friday asking whether Mueller felt Attorney General William Barr misrepresented their phone conversation in his Senate testimony this week.

Testifying before the Judiciary Committee on Wednesday, Barr said Mueller told him he did not think Barr's four-page summary of Mueller's conclusions was inaccurate, but rather that Mueller was unhappy with the media coverage of Barr's summary when he sent a letter objecting to the characterization of the special counsel report.

"Please inform the Committee if you would like to provide testimony regarding any misrepresentation by the attorney general of the substance of that phone call," Graham wrote.

Democrats have demanded that Graham bring in Mueller to testify himself, arguing that Mueller's March 27 letter shows Barr mischaracterized the special counsel report. Graham has been firm that he would not bring Mueller in to testify about the investigation, but he said at the end of Wednesday's hearing he would instead write to Mueller to give him the chance to respond.

"I'm going to write a letter to Mr. Mueller and I'm going to ask him, 'Is there anything you said about that conversation he disagrees with?' And if there is, he can come and tell us," Graham said at Wednesday's hearing.

The special counsel's office declined to comment.

It's not clear what Graham intended when he gave Mueller the opportunity to "provide testimony." Asked whether Graham has changed his position on allowing Mueller to testify publicly, Graham spokeswoman Taylor Reidy said: "He's giving Mueller the opportunity to provide testimony regarding any misrepresentation by the attorney general of the substance of that phone call. Individuals can provide testimony to a committee in a number of different ways."

Indeed, a witness can provide written testimony, meet with staff or lawmakers behind closed doors or testify in public, but the Graham spokeswoman would not clarify further.

In Mueller's letter, sent after Barr released his four-page summary and revealed this week, the special counsel wrote that Barr's characterization "did not fully capture the context, nature and substance" of the special counsel's investigation.

But Barr, who called Mueller's letter " a little bit snitty ," said Mueller's reaction was different in a follow-up phone conversation.

"I said, 'Bob, what's with the letter? Why didn't you just pick up the phone and call me if there's an issue?'" Barr said. "And he said that they were concerned about the way the media was playing this and felt that it was important to get out the summaries which they felt would put their work in proper context, and avoid some of the confusion that was emerging."

While Graham is not planning on having Mueller testify, the House Judiciary Committee is working to secure his appearance later this month, with May 15 as a target date, though it's yet to be finalized. Mueller is sure to be asked about his conversations with Barr if he appears there.

Graham interest in FISA allegations

Graham told reporters Thursday that he is moving forward on his own investigation into the Foreign Intelligence Surveillance Act and allegations that the FBI abused the surveillance law to obtain a warrant on Trump campaign adviser Carter Page.

Graham said he planned to speak about the matter with Chief Justice John Roberts, who appoints judges to the FISA court. He expressed concerns that Trump had soured on the program on the whole as a result of the accusations about the warrant for Page.

"I'm going to call Roberts," Graham said. "Oversight of the judiciary is part of our job, but I don't want to get into how a particular court operates. I don't intend to call the FISA judges. ... That is uncomfortable for me. But I am going to ask Roberts, will you please look and see what happened? Because I don't want to lose the FISA program."