Political operatives from New York to Washington, D.C., are buzzing about whether former New York Mayor Rudy Giuliani (R) is getting ready to launch a campaign against Sen. Kirsten Gillibrand (D-N.Y.) next year.

New York media outlets reported Thursday afternoon that Giuliani has decided not to run for governor in 2010 but that he is still mulling a Senate bid. A report in the New York Daily News, citing unnamed sources, said Giuliani had decided to run for Senate.

But a Giuliani spokeswoman said in a statement that no decision has been made.

“Rudy has a history of making up his own mind and has no problem speaking it,” spokeswoman Maria Comella said. “When Mayor Giuliani makes a decision about serving in public office, he will inform New Yorkers on his own.”

If he runs, this would be the second time Giuliani has sought this Senate seat. He mounted a brief campaign for what was then an open seat in 2000 but ended his campaign after being diagnosed with prostate cancer.