Former New York City Mayor Rudy Giuliani on Sunday did not rule out a run for the White House, but shed little light on his plans for the 2012 campaign.

During an appearance on CBS' “Face the Nation,” Giuliani told host Bob Schieffer he has mulled a run for the White House in “recent weeks.”

Since the Jan. 8 shooting in Arizona that claimed six lives and left Rep. Gabrielle Giffords (D-Ariz.) hospitalized, Giuliani said his thoughts have been on that situation.

Giuliani sought the Republican presidential nomination in 2008, but failed to gain much traction with votes during the GOP primary and eventually dropped out.