Jake Gyllenhaal, Taylor Swift Spend Weekend Together in NYC

Credit: Jon Kopaloff/FilmMagic; John Shearer/WireImage.com

New couple alert?

Jake Gyllenhaal and Taylor Swift spent the weekend together in NYC.

She first brought him to a Saturday Night Live taping.

Swift, 20, initially sat down with her bodyguard, but after the lights dimmed, he got up and Gyllenhaal, 29, occupied the seat.

Says a witness, "They chatted a lot, but no kissing or hand holding or anything affectionate."

The next day, they were spotted brunching and taking a stroll in Brooklyn's Park Slope neighborhood -- where Gyllenhaal's sister, Maggie, lives with her family.

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Still no official word from either star's rep.

If they are dating, Gyllenhaal better be careful how he treats her: She is known to write about her exes in songs.

"Every person I've fallen for, they've all been a song," she tells Us Weekly in Taylor Swift: Inside My World, a special collector's issue, which recently hit newsstands. She says she's "proud" of those songs. "That kind of justifies [the exes] being in my life."