The conversations with top aides have happened in recent weeks.
It's not clear if the issue has been discussed between the president and his advisers since the riots on Capitol Hill Wednesday. However, following the riots Trump's White House Counsel, Pat Cipollone, advised the president that he could face legal jeopardy for encouraging his supporters to storm the Capitol building, according to sources familiar with their discussions.
If a self-pardon happens, it's unclear when it would be announced. Trump is already expected to issue several pardons over the next two weeks.
News of a possible self-pardon was first reported by The New York Times.
Trump has long believed he has the power to pardon himself. In 2018 the president tweeted he had the "absolute right."
"As has been stated by numerous legal scholars, I have the absolute right to PARDON myself, but why would I do that when I have done nothing wrong? In the meantime, the never ending Witch Hunt, led by 13 very Angry and Conflicted Democrats (& others) continues into the mid-terms!" the president said in a June 2018 tweet referring to former Special Counsel Robert Mueller's investigation.
Although there is some ambiguity in the law, most constitutional lawyers and experts ABC News has spoken to say that a president cannot pardon him or herself, based on the founding legal principle that no person can be a judge or jury in their own case.
"A self-pardon clearly is inconsistent with that principle," said Mark Tushnet, a Harvard law professor.
Louis Seidman, a Georgetown professor who previously clerked for U.S. Supreme Court Justice Thurgood Marshall, said that a pardon "does imply something that one person gives to another, and it also runs up against obvious questions of conflict of interest."
But the Constitution does not address the issue, and some experts say this ambiguity leaves the door open.
"No president has ever tried to do that," said Seidman. "There is no precedent. I don't know of any Supreme Court cases that speak to it at all."
Jeff Powell, a Duke law professor who previously worked in the Department of Justice's Office of Legal Counsel, said some could use the lack of clarity in the Constitution, as well as previous Supreme Court rulings affirming "a broad power" of the presidential office, to argue that a president can pardon himself -- but those people would be on "the losing side of the argument."
Jonathan Turley, a George Washington University law professor who was a Republican witness during the House impeachment proceedings, wrote in a 2018 op-ed that it "does not make sense" that the framers of the Constitution would "[leave] this major limitation unstated."
"There is nothing to prevent Trump from adding his own name to the list of pardoned individuals," Turley said.
"There are lawyers who will defend that position," said Powell. "But most constitutional lawyers without a dog in the fight would come to the conclusion that he cannot."