Buzz is building around a possible White House run by Rudy Giuliani, who is considering a second bid for the GOP nomination.

New York Rep. Pete King (R) told Fox News last month that the former New York City mayor is "very close to running."

And the Weekly Standard's William Kristol reported Wednesday that Giuliani "may throw his hat in the ring soon."

"I'm told by two reliable sources that Rudy Giuliani intends to run for the GOP nomination for president in 2012," Kristol wrote.

An adviser to the Republican told The Ballot Box that Giuliani is indeed mulling a run, but that's the extent of it. "He is considering it, which doesn’t necessarily mean he is running," the adviser said.

Giuliani was considered a top contender for the GOP’s 2008 presidential nod, but his bid was derailed by a campaign strategy that focused on Florida and neglected Iowa, New Hampshire and South Carolina. He spent almost $50 million on his run, but failed to win any of the early states — his single delegate came from Nevada — and only finished third in Florida.

He later told Time magazine he regretted neglecting the early states.

"We had a great organization, we were really ready for a general election ... We had people in every state, we were organized in every state," he said. "So in retrospect I would have had a more focused campaign on Iowa and New Hampshire and South Carolina, and not built this organization – it was sort of like putting the cart before the horse."

Giuliani has since gone back to work at his international consulting firm, Giuliani Partners LLC. But he remains a prominent voice in the GOP and his standing when it comes to the issue of terrorism hasn’t diminished despite the 2008 electoral flop.

Should he decide to run, Giuliani would again face competition for the votes of the moderate wing of the party. Former Massachusetts and Minnesota governors Mitt Romney and Tim Pawlenty also appeal to moderate Republican primary voters.