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Pence made the remarks during a Q&A after a speech 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."
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.
Trump's pressure campaign against Pence
Two witnesses who advised Pence that he did not have the authority to subvert the election testified during the June 16 hearing: former Pence attorney Greg 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, Marc Short, Pence's chief of staff, 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.
"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 story has been updated.
CNN's Annie Grayer and Rachel Janfaza contributed to this report.