Washington (CNN) Special counsel Robert Mueller could testify before the House Judiciary Committee as early as next week, in what could become the most consequential congressional hearing of the Trump administration.

But it's not as simple as Mueller simply showing up to the committee room.

While Attorney General William Barr said last week that he has no objections to Mueller appearing before Congress, President Donald Trump reversed course and tweeted Sunday that Mueller should not testify.

Still, a hearing with Mueller remains on track to happen -- and soon. Here's what you should know about his potential upcoming testimony on Capitol Hill.

When is the hearing?

The House Judiciary Committee has floated May 15 as the date for a public hearing with Mueller, though the committee has not yet finalized the date.

But that might not be the only time the special counsel heads up to Capitol Hill to talk about his report and 22-month investigation. The House Intelligence Committee has requested that Mueller brief the panel on the counterintelligence investigation into Trump and Russia behind closed doors, though that has not been confirmed yet either.

It's also possible that Mueller could testify in the Senate. But it's not likely to be in the Senate Judiciary Committee. Chairman Lindsey Graham, a South Carolina Republican, said he had no interest in bringing in Mueller after he held a hearing on the report with Barr last week.

"For me, it's over," Graham said of the Mueller investigation.

Graham did add one possible exception, sending Mueller a letter last week and asking if he wanted to add any testimony specifically related to how Barr described their telephone call about Mueller's letter objecting to Barr's characterization the investigation.

Graham told CNN Monday that he's open to having public testimony from Mueller -- or any other form that Mueller wants to provide it -- about the Barr phone call. But Graham said that public testimony would just be about "any dispute" about the call.

Is Mueller allowed to testify?

Yes.

While Mueller has submitted his report, he still remains an employee of the Justice Department as the special counsel. And he's still driving in to his nondescript Washington office every day.

Congress has formally requested his testimony, and so long as the Justice Department doesn't object, Mueller can testify voluntarily. Barr was asked last week at a Senate hearing whether he had any issues with Mueller testifying, and Barr said he had no objections.

The format of the hearing has not yet been finalized either. It's possible that Mueller won't be the only one appearing from his team, and it's not clear whether he would be questioned just by lawmakers or also by staff attorneys -- as House Judiciary Chairman Jerry Nadler tried unsuccessfully to do with Barr last week.

What are the sticking points?

A potential wrench was thrown into Mueller's testimony on Sunday by Trump, who has claimed "total exoneration" from the Mueller report. The President had previously said he did not have a problem with Mueller testifying. But then he tweeted Sunday that Mueller should not testify.

"Bob Mueller should not testify. No redos for the Dems!" Trump tweeted.

An administration official familiar with the matter told CNN Monday night that Trump wasn't necessarily signaling his intent to block Mueller from testifying with that tweet.

"The President was expressing his opinion that the investigation is over and it's time to move on," the official said.

Trump himself could not stop Mueller from testifying, but he could try to convince his attorney general not to allow his testimony. If the Justice Department were to change course and oppose Mueller appearing before Congress, lawmakers would have several possible options to try to bring him in anyway.

The first would be to wait until Mueller is a private citizen, although the special counsel's office has given no indication when Mueller will leave the department.

The second would be to subpoena for his testimony.

In addition, the White House has another tool to try to limit Mueller's testimony: claim executive privilege over certain elements of the report. The White House waived executive privilege for the special counsel investigation and didn't try to block anything in the release of the report -- but the White House argues it has not waived executive privilege for Congress, and still reserves the right to invoke it with the legislative branch.

What is he expected to say?

It's impossible to predict what Mueller will say. But his background as a straight-laced Marine who never veers from the rules gives us some clues about how this all might play out.

Mueller will probably try to hug his report as closely as possible. He probably knows it better than any of the lawmakers will, anyway. So, when he is asked about the most explosive episodes in the report, Mueller could easily answer by pointing to what's in the report. This is the safe option for Mueller, who can rely on the established record and try to stay in bounds.

But there will be other angles where Mueller might be boxed in. What was his reaction to Barr's press conference? Does he think Barr mischaracterized his findings? Would he have sought an indictment if the target of the investigation was not the sitting President? Did he intend for Congress to pick up the ball where he left it? It'll be up to Mueller to decide how to answer.

He could make a lot of news with his answers. Depending on what he says, the public's entire interpretation of the results of the investigation could change. Or, his testimony could be a huge letdown, where nothing new is learned and Mueller gives staccato answers to poorly worded questions.

Could he testify about some things in private?

Absolutely. There aren't any closed-door hearings on the books yet, but that could come soon.

House Democrats would be smart to proactively schedule an open session and a private session with Mueller, because he won't be able to speak publicly about everything in the report. There are redacted sections that contain classified information about Russian cyberattacks.

Other parts touch on ongoing mattes that Mueller won't want to discuss publicly, like how Trump ally Roger Stone tried to backchannel with WikiLeaks on behalf of the campaign. But with Stone's trial this fall, questions about his case will probably be forced behind closed doors.

Already, the top Democrat and Republican on the House Intelligence Committee, who don't agree on much, have jointly asked Mueller for a private briefing. So, it's a very safe bet that Mueller will be spending time this year in closed-door sessions on Capitol Hill.