Appearing on stage with him at an event on global leadership at Vanderbilt University in Nashville, Tennessee, she told him: "I can't imagine declining to testify. I would feel like I was violating the oath that I took to the Constitution – to defend the country against all enemies foreign and domestic."
Mr Bolton himself served as national security adviser to Mr Trump before being dismissed in September.
During the president's impeachment trial in the Senate on charges of abuse of power and obstruction of Congress, it was reported that Mr Bolton's as yet unpublished book contained information suggesting he had crucial information – that Mr Trump had specifically told him that he was withholding military aid to Ukraine until officials in Kiev announced investigations into his political opponents.
A vote in the Senate on demanding Mr Bolton give testimony was defeated by Mr Trump's Republican allies. He had said he would testify if ordered to.
However, Dr Rice insisted he should have done so even without efforts to force him to.
She said: "I can't imagine withholding my testimony, with or without a subpoena.
"I also can't imagine, frankly, in the absence of being able to provide that information directly to Congress, not having exercised my First Amendment right to speak publicly at a time when my testimony or my experience would be relevant."
She said that she had agreed to testify at the Benghazi hearings without a subpoena, even though she did not want to take time away from her duties as national security adviser, a role she held under Barack Obama from 2013 to 2017.
However, Mr Bolton insisted he had done nothing wrong – and called the impeachment hearings " grossly partisan".
He said: "People can argue about what I should have said and what I should have done.
"I would bet you a dollar right here and now, my testimony would have made no difference to the ultimate outcome."
Mr Bolton says he hopes to publish his book next month and says he does not believe it contains the kind of classified information that could see it withheld.
Democrats in the House of Representatives have not ruled out calling on him to testify, even though the impeachment charges they sent to the Senate have already been defeated.