Thanks to Rudy Giuliani and Mike Bloomberg, the Republican Party has grown accustomed to being a player in the NYC mayor’s race.

With 2013 right around the corner and several Democrats jockeying for position in hopes of succeeding Bloomberg, there’s no clear GOP contender in sight.

Several possibilities have been mentioned: NYPD Commissioner Ray Kelly, who hasn’t yet shown any interest in running; and Democrat-turned-Republican supermarket mogul John Catsimatidis, who gave up his mayoral aspirations in 2009 when Bloomberg successfully pushed to extend term limits and ran again himself.

Catsimatidis, who is close to state GOP Chairman Ed Cox thanks to the marriage of their respective children last summer, has said he’ll only run if no one else does.

Just last week, long-shot Democratic contender Tom Allon confirmed to the Wall Street Journal that he’s mulling a Democrat-to-Republican switch – a move that would mimic Bloomberg’s pre-2001 campaign party hopping. (The mayor, of course, is now a man without a party, but managed to convince – with the help of some generous campaign contributions – the five NYC GOP county leaders to let him run on their ballot line yet again in ’09).

Brooklyn GOP Chairman Craig Eaton has floated Sen. Marty Golden’s name as a potential GOP contender – not the first time that has come up.

Now comes word – confirmed by multiple GOP sources – that the Republicans are interested in finding a woman to run, perhaps thinking that’s their best bet to go up against the Democratic frontrunner, NYC Council Speaker Christine Quinn.

The GOP is so intent on locating a female candidate for mayor, that at least one well-connected Democrat has been approached to help the party suss out a potential contender.

One name being circulated is Eva Moskowitz, a former NYC councilwoman and head of a fast-growing – and controversial – chain of charter schools. Another name also comes from the education field: Merryl Tisch, chancellor of the New York State Board of Regents, but she has repeatedly insisted she’s not interested in the job.

A top Republican said the party hasn’t ruled out cross-endorsing a “really great” Democratic candidate, should one come along.

This source did not believe any of the current crop of Democratic candidates falls into that category, but did suggest that a well-known Democrat – “an Adolfo Carrion, or Harold Ford Jr.” (neither of whom has yet indicated any interest in running) – could yet leap-frog over that group and land the support of the GOP, assuming, of course, that he would want it.

The goal, the source said, is “making New York City gets the mayor it deserves” – by whatever means necessary.