(CNN) - Jeb Bush, whose endorsement of Mitt Romney helped secure the candidate as the all-but-certain GOP nominee, said an interview Romney should avoid the urge to wage a negative campaign. The popular former Florida governor also offered up his pick for Romney's running mate, and said he'd consider a spot on the GOP ticket if asked.
"I think Mitt needs to stay above the fray a bit, and to offer a hopeful message that can lift people's spirits up," Bush said in an interview this week with the conservative website Newsmax. Video of the interview was posted on Newsmax's website.
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Bush said American voters have had their fill of nasty campaigns after a blistering GOP primary process.
"After the end of this four or five months of really negative campaigning I think people are going to be motivated by a more positive message," Bush said, adding that he believed personal attacks should stay out of the campaign.
"I don't think we need to demonize the president," Bush said. "I know a lot of people when my brother was president spent a lot of energy trying to demonize a good honorable man."
"That's not what campaigns ought to be about," he added.
Bush served as Florida's governor from 1999-2007, and is the son of former President George H.W. Bush. His brother is former President George W. Bush.
Jeb Bush was considered a favorite to jump into the 2012 GOP contest, but stated firmly in May 2011 he would not be a candidate for his party's presidential nomination. He declined to endorse a candidate ahead of Florida's January 31 primary, saying he wanted to allow a candidate to "earn" his victory in the state.
Jeb Bush's name is often mentioned as a potential running mate for Romney, though the former Florida governor downplayed any speculation in the interview.
"Well I'd consider it," Bush said of joining a GOP presidential ticket. "I doubt I'll get a call, and I don't know if it's the right thing for me to do. I didn't run for president for a similar kind of reason, so I'm all in to try to help him get elected."
Bush did, once again, offer up another name as a strong choice for Romney's running mate – the junior senator from his state, Marco Rubio.
"Well I can't speak for Gov. Romney, and I can't speak for Sen. Rubio, but if I was on both sides of that conversation I would ask – and I would hope - that Marco would accept," Bush said.
Bush said he thought "the combination would be extraordinary" between Romney and Rubio. Last month Bush also hailed Rubio as a strong vice presidential pick, saying the tea party favorite was "the best orator of American politics today, a good family man."
Rubio has consistently indicated he would turn down a spot on the GOP ticket if asked.