Earlier this week, strategist Karl Rove attacked Clinton's health, suggesting she suffered traumatic brain injury from a 2012 fall.
Her husband, former president Bill Clinton, insisted she works out regularly and is in "better shape" than him. And while he said there was nothing to Rove's charge, the ex-president acknowledged a candidate's health is fair game in national campaigns.
In addition, Clinton has been under attack for her handling of the 2012 terror attacks in Benghazi, Libya and on her record on the terrorist organization Boko Haram, which kidnapped nearly 300 Nigerian teenaged girls from their school and threatening to sell them.
The State Department, under Clinton, refused to designate Boko Haram — which is affiliated with al-Qaida — as a terror organization. That designation was not made until last November, well after Clinton had resigned as secretary of state.
Clinton's age has also come under fire. She will turn 69 years old two weeks before the 2016 election. Late President Ronald Reagan, in comparison, was eight months older than that when he ran for office in 1980.
The former first lady's history will be an issue should she decide to run, said Priebus Sunday.
"I think Hillary is a known product," he said. "Actually, I think it's sometimes worse running against a blank slate. Hillary has decades of history for us to explore."
Priebus said her role in "Hillarycare," her push for health insurance reform while she was still first lady, will be brought up, as will her experience as a New York senator, "where there's nothing significant to point to."
Further, Priebus said, Clinton's experience as secretary of state "is just not significant, but there's all kinds of problems with her."
And when she was asked questions about Benghazi and her response was "what difference does it make," Priebus said, "I assure you it will be an issue if she decides to run."
But Priebus said that despite calls for Rove to apologize for his comments, that will be "up to Karl Rove. He's a political operative."
But he thinks health and age is "fair game."
"It was fair game for Ronald Reagan. It's fair game with John McCain," said Priebus, noting that people attacked the Arizona Republican, saying he was "maybe psychologically not fit because he was a prisoner of war."
But he's not sure there is "a graceful way to bring up age, health, and fitness for a candidate that wants to be president of the United States. I think the more important issue for me as leader of this party is what's the record of Hillary Clinton? What was her record as a secretary of state, Benghazi, Boko Haram, Syria. Those are the issues."
Priebus, meanwhile, refused to speculate on Rove's claims.
"I'm not a doctor," Priebus said. "What I do know is the issue will come up as it does for any person running for president. What I think is going to make her rethink whether she should actually run for president... She's coming out with a book called "Hard Choices" or something like that. She's made a series of bad choices."
He also accused Clinton of trying to sweep Benghazi under the rug.
"If you want any evidence ask the families of people who lost their sons in Benghazi," he said. "They've talked plenty about what happened in Benghazi."
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