Drew, who split from Strokes drummer Fabrizio Moretti in January, was caught canoodling with Spike during dinner at Santa Monica's Ivy at the Shore restaurant on Tuesday.
A source told website X17 online: "They were all over each other during their date."advertisement
Spike, 37, who directed 'Being John Malkovich' starring Drew's close friend Cameron Diaz, is the ex-husband of director Sofia Coppola.
Sofia was reportedly so irritated by Cameron when she worked with Spike, she based the obnoxious, dizzy blonde character of Kelly in 'Lost In Translation' on her.
Meanwhile, Drew has confessed she can't stop smiling and is "disgustingly" optimistic.
The 32-year-old said: "I don't know why I am always this happy but I have such a great life. I can't help but be happy. I am just disgustingly always optimistic. It's got to be weird."
Drew also revealed she is much happier since splitting from Fabrizio, who she dated for four-and-a-half years, because she can channel her energies into making herself happy.
She admitted: "It's a win-win situation."