Originally Published: 17 AUG 22 11:01 ET

Updated: 17 AUG 22 13:32 ET

By Athena Jones, CNN

(CNN) -- Former Vice President Mike Pence said Wednesday he would give "due consideration" to any formal invitation to testify before the House select committee investigating the January 6, 2021, attack on the US Capitol, while hinting at potential executive privilege issues.

Pence made the remarks during a Q&A after a speech here at the New Hampshire Institute of Politics' "Politics & Eggs" breakfast, a common stop for candidates considering a run for office.

"If there was an invitation to participate, I would consider it," Pence said, after calling January 6 a tragic day for all Americans. "But, you heard me mention the Constitution a few times this morning. Under the Constitution, we have three co-equal branches of government, and any invitation to be directed to me, I would have to reflect on the unique role I was serving in as vice president. It would be unprecedented in history for a vice president to be summoned to testify on Capitol Hill. But, as I said, I don't want to pre-judge, so if there's ever any formal invitation rendered to us, we would give it due consideration."

While Pence said it would be "unprecedented" for a vice president to be asked to testify on Capitol Hill, presidents and vice presidents have testified before Congress in the past.

They include President Gerald Ford, who testified voluntarily in front of the House Judiciary Subcommittee on Criminal Justice in 1974 to explain why he pardoned his predecessor Richard Nixon.

Serious constitutional issues

A person with knowledge of Pence's thinking cautioned against reading too much into his remarks Wednesday.

The source noted that the former vice president would have serious constitutional issues about appearing before the January 6 panel, citing Pence's comments about the branches of government being co-equal.

Pence also believes that much of the information related to his experience on January 6 has been provided to the committee, given that his former chief of staff, Marc Short, and Pence lawyer Greg Jacob have already testified in full, the source said.

The January 6 committee had no comment.

The select committee detailed at a public hearing in June how former President Donald Trump tried to pressure Pence, his vice president, to join in his scheme to overturn the 2020 presidential election -- and how Pence's refusal put his life in danger as rioters called for his hanging on January 6, 2021.

Two witnesses who advised Pence that he did not have the authority to subvert the election testified during the June 16 hearing: Jacob and retired Republican judge J. Michael Luttig.

The committee walked through how conservative Trump attorney John Eastman put forward a legal theory that Pence could unilaterally block certification of the election -- a theory that was roundly rejected by Trump's White House attorneys and Pence's team but nevertheless was embraced by the former President.

In a videotaped deposition, which was played on June 16, Short said the vice president advised Trump "many times" that he didn't have the legal or constitutional authority to overturn the results while presiding over the joint session of Congress on January 6 to count the electoral votes.

A call for 'unprecedented transparency'

On Wednesday, Pence also reiterated his call for US Attorney General Merrick Garland to give the American people a full accounting of the reasons the recent search of Trump's Mar-a-Lago resort was necessary, citing the politicization of the FBI during the Trump administration.

"This unprecedented action does demand unprecedented transparency," Pence said, while going on to defend the rank-and-file agents. "I just want to remind my fellow Republicans, we can hold the AG accountable for the decision he made without attacking rank-and-file law enforcement personnel at the FBI. The Republican Party is the party of law and order. Our party stands with the men and women who serve on the thin blue line at the federal and state and local level, and these attacks on the FBI must stop. Calls to defund the FBI are just as wrong as calls to defund the police."

Pence spoke before a packed room, with about a dozen people standing along the walls. As he has done in recent speeches, he argued that elections are about the future and not the past and that the GOP must offer a positive vision for the future in order to win.

This was the latest of several trips Pence has made to states with early presidential nominating contests; New Hampshire holds the first-in-the-nation presidential primary, a key contest in the race for the White House. After the speech, he was set to spend the day campaigning with candidates for state Senate and participating in roundtables with business leaders and law enforcement officials before delivering a keynote address at a Tri-County GOP event.

This story has been updated.