Romney's list of possible VP picks includes SASC member Ayotte

"There are terrific Republicans in the Senate, in the House, in governors' offices," Romney said on Fox News. "You've got extraordinary men and women. We have a very deep bench."

Ayotte endorsed Romney over the weekend, and serves as co-chair of Romney's national advisory committee.

Romney has demurred in the past when asked about his prospective running mate, and initially did so with Hannity, saying it would be “too presumptuous” to name names ahead of the nomination.

“I hope to become the nominee and then we'll start thinking about things like that,” Romney said. "Whoever our nominee is will be able to choose a remarkable, superb person to become the vice presidential contender."

Romney released his first attack ad against President Obama, timed to confront the president's visit to New Hampshire on Tuesday.

"We have to have a new president. I hope I'm that person," Romney said. "I think I've got the best shot at defeating President Obama."