After dates with BILLY JOEL’s EX, RUSSELL CROWE streaks back home to his cheating wife!

 EVEN though Russell Crowe was spotted getting cozy with Billy Joel’s ex-wife Katie Lee in two New York hotspots recently, the Aussie hunk is now talking about patching things up with his estranged wife Danielle Spencer!

“Russell desperately wants to recon­cile with Danielle,” said a source close to the couple. “He’s begging her to give their marriage another go.”

But it sure didn’t look that way in early December when Russell, 48, and cute foodie Katie, 31, were seen cuddling in the Drawing Room of the Greenwich Hotel in Manhattan on Dec. 9.

Katie, who hosted the first season of TV’s “Top Chef ” before writing her own cookbook, was also at Joe’s Pub on Dec. 8 to watch Crowe perform with an all-star crowd.

Suddenly, Russell seems to have had a change of heart. The Oscar winner – who’s been shooting “A Winter’s Tale” in New York between appearances at worldwide premieres of “Les Misera­bles” – wrote on his Twitter page: “As soon as I finish this job and can get home, my priority is to try to bring my family back together.”

As The ENQUIRER reported  in October, Russell never got over Dani­elle, 42, stepping out with Damian Whitewood, her hunky partner on the Down Under version of “Dancing with the Stars” in late June, sources say.

“Russell’s ego couldn’t handle it,” said a source.

But now, sources say he realizes that a divorce would hit him where it hurts the most – in the pocketbook!

His real estate holdings alone are estimated at $50 million, meaning Danielle could walk away with tens of millions.

Crowe has already tried to scuttle rumors of his marital split by planting a passionate kiss on Danielle when she arrived in L.A. with their boys Charles, 9, and Tennyson, 6, a few days after the scandal broke.

And now he’s flatly denying that there was anything going on between him and Katie.

“There was obviously an attraction,” said a source. “But now Russell’s focus­ing all of his attention on Danielle.”

AND he was seen in Sydney, Australia this morning visiting the home of Danielle, where she and their children live.

Reportedly, he got out of the car laden with a big bag of Xmas gifts!