Bolton repeatedly deferred to say whether he would take the job, saying his priority was simply "replacing President Obama."
"I think once that objective is achieved, then we can play the game of who is going to be in the Cabinet and who is going to have senior positions," Bolton said.
Bolton did praise Gingrich for having "pretty good insight" on national security questions, although declined to endorse any of the GOP candidates.
"I haven't endorsed anybody. I'm still happy to be talking to a number of the candidates and happy to provide whatever support I can to all of them at this point," Bolton said.
He also said that a Republican candidate needed to do a better job at articulating a clear foreign policy.
"President Obama appears to be poised to make as his principle argument that under his administration a number of terrorists have been killed and that that ends the foreign policy debates," Bolton said. "I certainly congratulate him on those successes but that doesn't make a foreign policy. And the challenges that we face and our friends and allies face around the world are growing and we are not responding to them adequately. Whoever our nominee is has to be prepared to engage in that debate."