Trump's personal lawyer Rudy Giuliani speaks during a press conference at the Republican National Committee headquarters in Washington, DC, on November 19, 2020.
CNN  — 

An attorney for Rudy Giuliani said on Thursday he’s concerned the House select committee investigating January 6 won’t respect his client’s privilege claims after the panel asked a judge to waive attorney-client privilege in the case of another Trump lawyer.

That concern makes the possibility of Giuliani cooperating with the committee’s investigation even more remote. There had been ongoing negotiations between Giuliani’s legal team and the committee, but no formal cooperation agreement has been reached.

The committee argued in an explosive court filing on Wednesday night that right-wing lawyer John Eastman’s privilege claims should be waived because he and former President Donald Trump engaged in a “criminal conspiracy” to overturn the 2020 presidential election.

Giuliani’s lawyer, Robert Costello, said in an interview with CNN that the committee’s attempt to get around attorney-client privilege in the Eastman case could throw a wrench into his talks with them. Costello said the committee “has some explaining to do.”

“I am going to have to have a conversation with these people about his attorney-client privilege, work product privilege and executive privilege,” Costello said. “We were operating under the idea that they would respect the invocation of those privileges, but they clearly haven’t respected the invocation of attorney-client privilege for Eastman. And so, if they didn’t do it with him, why would they do it with respect to Rudy Giuliani?”

RELATED: Takeaways from the House’s January 6 committee’s claims of a potential Trump ‘criminal conspiracy’

Giuliani was subpoenaed by the committee in January and has been engaging with lawmakers, through his lawyer, about the scope of the subpoena and whether he may be able to comply with some requests. CNN previously reported that Giuliani may be willing to testify about claims of election fraud but that he did not intend to waive executive- or attorney-client privilege.

The committee previously said it still expects Giuliani, a central figure in Trump’s failed bid to overturn the 2020 election, to “cooperate fully” with its subpoena.

Wednesday’s filing is the most extensive release to date from the House’s January 6 investigators. House members have previously signaled they may make a criminal referral to the Justice Department about Trump, depending on their findings, and the House’s arguments Wednesday could be seen as a preview of a case that could be made by federal prosecutors.

Democratic Rep. Elaine Luria of Virginia told CNN that the committee’s filing on Eastman should serve as a warning for others who do not want to cooperate, like Giuliani.

“It certainly should be,” Luria said, when asked if the panel’s handling of Eastman’s claims of privilege should affect others who are making that same claim.

Referencing Giuliani, who has filed his own privilege claim, Luria said, “I think hearing from these people directly would be much to their own benefit, so that they can tell their side of the story.”

CNN’s Annie Grayer contributed to this report.