(CNN) Attorney General William Barr has warned Democrats on the House Judiciary Committee he won't show up to this week's highly anticipated hearing if they stick to the format the chairman has proposed for the questioning, according to a committee source with knowledge of the matter.

Skipping this week's hearing would amount to a dramatic escalation in the growing fight between the Trump administration and House Democrats over a range of oversight requests, including over access to the unredacted report from special counsel Robert Mueller, the subject of Thursday's hearing.

But it's still possible they could reach a deal by Thursday, meaning Barr might ultimately show.

House Judiciary Chairman Jerry Nadler told CNN on Sunday morning that Barr would not "dictate the format of the Judiciary Committee."

"The witness is not going to tell the committee how to conduct its hearing, period," Nadler, a New York Democrat, said.

Asked what he would do if Barr doesn't comply, Nadler said: "Then we will have to subpoena him, and we will have to use whatever means we can to enforce the subpoena."

Nadler, wants to allow all members of his panel at Thursday's hearing to have one round of questioning of five minutes each, according to the source. He also wants to allow for a subsequent round of questioning of 30 minutes for each side, allowing both parties' committee counsels to also engage in questioning during their respective turns — which has turned into a key sticking point for the Justice Department.

"The attorney general agreed to appear before Congress. Therefore, members of Congress should be the ones doing the questioning," said Justice Department spokeswoman Kerri Kupec. "He remains happy to engage with members on their questions regarding the Mueller report."

Nadler also is proposing that the committee should go into closed session to discuss the sections of the report that are redacted.

But Barr has rejected those proposals for additional rounds of questioning, according to the source. The Justice Department has informed Nadler's office that Barr doesn't think the committee counsels should be allowed to question him, the source said, prompting the attorney general to threaten to not show up next week if Nadler follows this format, the source said.

A separate source said that the Justice Department made it clear in negotiations: their position is that it's a congressional hearing, and so only members should do the questioning.

Barr also has objected to holding a closed session to discuss the full report.

It's unusual — but not unprecedented — to allow committee counsels to question witnesses. But doing so would allow the subject-matter experts to follow up on questions the lawmakers may have missed from earlier rounds. Barr has proposed allowing a small group of lawmakers to review a less-redacted report, but House Democrats have issued a subpoena for the full report.

A spokesperson for the House Judiciary Committee Republicans criticized Nadler's fight with Barr over the format after the attorney general offered to testify voluntarily.

"The only thing, apparently, that will satisfy Democrats, who refuse to read the less-redacted report, is to have staff pinch hit when a cabinet official appears before us," the spokesperson said. "What actual precedent is there for our committee making such demands of a sitting attorney general as part of our oversight duties? The attorney general isn't a fact witness, and this committee's investigations — as Democrat leadership reminds us daily — don't constitute impeachment, so Democrats have yet to prove their demands anything but abusive and illogical in light of the transparency and good faith the attorney general has shown our committee."

Barr is scheduled to testify before the Senate Judiciary Committee on Wednesday and the House Judiciary Committee on Thursday. And he's still expected to attend the Senate hearing.

While Senate Judiciary Chairman Lindsey Graham, a South Carolina Republican, has backed Barr's handling of the Mueller report, the quarrel over the testimony is the latest in a series of escalating disputes between Nadler and Barr over the special counsel investigation.

Nadler has issued a subpoena for the unredacted Mueller report and the special counsel's underlying evidence, and is prepared to go to court to obtain the documents if Barr will not comply. Barr has offered to allow select congressional leaders to view a less-redacted version of the Mueller report, with only grand jury material blacked out, but Democrats have rejected that proposal.

But the dispute between Barr and Democrats runs deeper than the report. Democrats have slammed Barr's handling of the end of the Mueller investigation, accusing him of releasing a letter that mischaracterized Mueller's findings on both collusion and obstruction and raising suspicions over his decision not to prosecute Trump after Mueller did not reach a conclusion on Trump. Barr's decision to hold a press conference the morning before the report was released was seen by Democrats as a blatant attempt to spin the report in defense of the President.

Barr's scheduled testimony comes ahead of House Democratic plans for Mueller to also appear next month to discuss the report and investigation. If Barr does not appear before the House Judiciary Committee, it's not clear how that would affect the panel's plans for Mueller's testimony. Graham has said he does not want to hear from Mueller himself because it would be too much of a circus.