(CNN)Rudy Giuliani has been engaging with the House select committee investigating the January 6 insurrection through his lawyer about the scope of his subpoena and whether he may be able to comply with some requests.

       The panel is making clear that it still expects Giuliani, a central figure in former President Donald Trump's failed bid to overturn the 2020 election, to "cooperate fully" with its subpoena. CNN reported last week that Giuliani was among four witnesses scheduled to appear before the committee on Tuesday who had their depositions rescheduled.

       A running list of who the January 6 committee has subpoenaed or requested to appear

       "Mr. Giuliani's appearance was rescheduled at his request. He remains under subpoena and the Select Committee expects him to cooperate fully," a committee aide said in a statement to CNN.

       Robert Costello, Giuliani's attorney, previously told CNN that the committee deferred a subpoena deadline while discussions are ongoing. The committee has rescheduled interviews for other high-profile witnesses who have engaged with investigators, but those talks are predicated on negotiations happening in good faith. Costello has so far declined to say in what areas Giuliani may be willing to cooperate.

       A source familiar with the matter told CNN that Giuliani may be willing to testify to the committee about claims of election fraud after the 2020 presidential contest, though the former Trump attorney still does not intend to waive executive or attorney-client privilege. More than a dozen items of interest to the committee in Giuliani's subpoena have to do with election fraud, and Giuliani may be willing to testify under oath about some of those matters as they are not covered by privilege, the source said.

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       But negotiations are still in the early stages, and there is still no agreement for Giuliani to cooperate. And he hasn't turned over any documents to the committee, the source said.

       The New York Times reported Saturday that Giuliani's lawyer is discussing with the committee the possibility of responding to questions and "has signaled to the committee that he plans to take a less confrontational stance toward its requests than some other members of Mr. Trump's inner circle who are fighting the committee's subpoenas or have otherwise refused to cooperate." The Times also reported Giuliani is still negotiating over whether to sit for a formal deposition or give an informal interview.

       Giuliani's apparent willingness to engage with the committee comes as a stark shift from his previous stance.

       Last month, immediately following news he had been issued a subpoena, CNN reported that Giuliani had indicated through his attorney that he didn't plan to provide information to the committee, citing claims of executive privilege and attorney-client privilege.

       GOP Rep. Adam Kinzinger said on Sunday that committee members "fully expect" Giuliani to speak with them, adding, "The expectation is he is going to cooperate because that's the law."

       "Regardless of when we hear from Rudy or how long that interview is, we're getting a lot of information, and we're looking forward to wrapping this up at some point, when that is right, showing it to the American people, but not rushing it, not hurrying this. We want everyone to have the full story," Kinzinger said on CBS' "Face the Nation."

       Former White House chief of staff Staff Mark Meadows was referred to the Justice Department for possible criminal contempt charges after refusing to comply with his subpoena in December. Steve Bannon was indicted in November after refusing to obey his subpoena.

       This story and headline have been updated with additional developments Monday.