Asked by Justice Elena Kagan what "core" executive functions have "absolute protection," government attorney Michael Dreeben said they include pardon power, veto, foreign recognition and appointments.

"Congress cannot say you can't appoint a federal judge who hasn't received a certain diploma, hasn't achieved a certain age," he said.

Commander in chief is also on the list, he said, though added that Congress has "substantial authority in the national security realm."

"I think that there may be some aspects like directing troops on the field in which the president's power is completely unreviewable," he said.

As the justices wrestle with immunity, they are posing another question to lawyers: Can presidents pardon themselves?

Justice Neil Gorsuch asked Trump attorney John Sauer about the possibility, saying presidents could be incentivized to do so if they fear their successors could prosecute them for actions they took while in office.

"I didn't think of that until your honor asked it," Sauer said. "That is certainly incentive that might be created."

Michael Dreeben, arguing for the government, was later asked for his view on whether the president has such authority.

"I don't believe the Department of Justice has taken a position," Dreeben said. "The only authority that I'm aware of is a member of the Office of Legal Counsel wrote on a memorandum that there is no self-pardon authority. As far as I know, the department has not addressed it further, and the court had not addressed it either."

Justice Samuel Alito addressed the "layers of protection" against bad faith prosecutions raised by the DOJ.

"I'm going to start with what the D.C. Circuit said. So, the first layer of protection is that attorneys general and other justice department attorneys can be trusted to act in a professional and ethical manner, right?" Alito said. "How robust is that protection?"

Alito continued that the "vast majority" of attorneys general and justice department attorneys "take their professional ethical responsibilities seriously" but that there have been exceptions.

In response, DOJ attorney Michael Dreeben agreed that there have been "rare exceptions" and said that "we're talking about layers of protection."

"I do think this is the starting point, and if the court has concerns about the robustness of it, I would suggest looking at the charges in this case," he said.

"The allegations about the misuse of the Department of Justice to perpetuate election fraud show exactly how the Department of Justice functions in the way that it is supposed to," Dreeben said. "Petitioner is alleged to have tried to get the Department of Justice to send fraudulent letters to the states to get them to reverse electoral results."

Michael Dreeben, arguing for Smith's team, faced questions from Justice Samuel Alito on whether or not presidents can make a "mistake" given the many competing pressures they are under in their day to day duties.

"Presidents have to make a lot of tough decisions about enforcing the law and they have to make decisions about questions that are unsettled," Alito said, then asking if a "mistake" makes a commander in chief criminally liable.

"Making a mistake is not what lands you in a criminal prosecution," Dreeben said.

Later he raised some of the specific accusations in the charges against Trump: "It is difficult for me to understand how there could be a serious constitutional question about saying, 'You can't use fraud to defeat the [certification of the winner of the presidential election], you can't obstruct it through deception, you can't deprive millions of voters of their right to have their vote counted for the candidate who they chose.'"

As Smith's team has noted in arguing against Trump, the former president's lawyers during his second Senate impeachment trial said in very clear terms that they believed a president could in fact be criminally prosecuted by the Department of Justice -- the opposite of what his legal team is currently arguing.

"If my colleagues on this side of the chamber actually think that President Trump committed a criminal offense -- and let's understand, a high crime is a felony and a misdemeanor is a misdemeanor; the words haven't changed that much over time -- after he is out of office, you go and arrest him," one of his attorneys said in opening statements during his impeachment trial in 2021, shortly after Jan. 6.

"We have a judicial process in this country, we have an investigative process in this country, to which no former officeholder is immune,” Trump's attorney said then.

But now, facing federal charges related to Jan. 6, Trump's lawyers argue a president can only be prosecuted if he is impeached, convicted by the Senate and removed from office.

The Senate in his second impeachment trial acquitted, though a majority of lawmakers voted to convict him.

-ABC News' Katherine Faulders