Huntsman told NBC's "Meet the Press" that the information needs to come out "in total" and that the matter has distracted from real issues on the campaign trail.
"Legitimate questions have been raised and that information has to come forward," said Huntsman, who added that it is up to Cain to divulge the details.
"This is taking all the bandwidth out of the discussion," he said. "So we're not able to talk about jobs. We're not able to talk about our position in the world. That hurts the American people."
Cain has repeatedly denied ever sexually harassing anyone.
He told reporters Saturday night that he won't answer any more questions about the decade-old allegations from at least three women.
J.D. Gordon, a spokesman for the Cain campaign, said Sunday that Cain looked forward to getting back to focusing on the country's major issues, such as the economy and national security. He blamed a "malicious smear campaign" by the media for steering attention elsewhere.
But Huntsman isn't the only Republican calling for Cain to set the record straight.
Mississippi Gov. Haley Barbour, also appearing on NBC, said the best way for Cain to get back on message "is to get all the facts on the table."
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