The pair first met at the Kids' Choice Awards back in March after being introduced by Justin Bieber. They reportedly dated briefly, before going their separate ways after Styles was snapped smooching with American model Emma Ostilly in Auckland back in April.
Swift went on to enjoy a whirlwind romance with 18-year-old Conor Kennedy, who is the son of Robert F Kennedy Jr, before calling it quits a few weeks ago.
Added another source: "Harry turned up at rehearsals to watch Taylor. She delayed it 20 minutes to wait for him. He was sitting with her mum Andrea and she seemed won over too. After Taylor finished, he picked her up over his shoulder and carried her to her trailer.
"They were holding hands and flirting all day. They had matching chains on and queued up together to get their lunch from the catering truck. It was all really sweet."
'OFFICIALLY HANGING OUT'
X Factor co-host Mario Lopez also dished on his radio show on Friday that he saw them "walking off hand in hand" the previous day.
"Taylor Swift was the special guest performer, and [here's] a little inside scoop for you," Lopez told his listeners.
"During rehearsals, Harry from One Direction came and slapped me on the back, and said, 'Hey, Mario, how ya doing?' And I said, 'What are you doing here?' And he sort of [pointed] toward Taylor."
Lopez went on to say he later saw the two "walking off hand in hand. So Taylor Swift and Harry from One Direction - you heard it here first. [They're] officially hanging out, I can say that much."
Listen to Lopez discuss the pair's backstage PDA here.
E! News also confirms that Styles was spied hanging out with Swift on the show's set last week.
"He watched her onstage going through her performance. He was sitting with her mom and the rest of her entourage," shares a source.
"After she finished, he went up to the stage. They were joking and laughing. He picked her up and threw her over his shoulder and carried her off."
Meanwhile, an 'insider' tells scandal sheet Radar Online that Swift and Styles are giving romance another shot.
"Taylor and Harry have been quietly seeing each other again," says the source.
"When Taylor dates someone she jumps in full force, but Harry is being a little more cautious this time around.
"Harry's being really sweet with her right now and careful not to hurt her. But at the same time, he's a young, popular heartthrob who isn't quite ready to be tied down."
Another source dished to The Sun: "She thought Harry was hot and got Justin [Bieber] to introduce them. She fell for him straight away. They secretly had a bit of a thing.
"They were texting and talking a lot and met up a few times. But it ended when Harry was spotted with another girl in New Zealand.
"Taylor was really embarrassed by what happened and was nervous about giving him another go. But this time they have spent more time getting to know each other."
They added: "Taylor is very full-on in relationships and falls for people quickly. Harry is being careful because he doesn't want to hurt her again.
"But he's made it clear he likes her a lot. They have been quietly seeing each other again for a little while now."
'IF WE BREAK UP, I'M GOING TO WRITE ABOUT IT'
Swift, whose love life provides endless inspiration for her hit songs, also reveals to The New York Times Magazine that she once told a boyfriend she wouldn't write a song about him - but she went ahead and did it anyway.
Asked by the mag if her boyfriends ever say "Look, whatever happens, just don't write about me," Swift replied, "The only time that has ever crossed someone's mind was when we were in the process of breaking up."
She added: "Of course I was like, 'Oh, don't worry, I won't'. And then I did.
"Look, it's not like it was written somewhere in the fine print that I write songs about my life. If we break up, I'm going to write about it. But I'm probably also going to write about when I fell in love with you. So there's an upside."
She also declined to confirm whether her split from Gyllenhaal inspired her new album, Red.
"I don't talk about who it is specifically because these are real people. You try to give insight as to where you were coming from as a writer without completely throwing somebody under the bus," she said.
Check out her full interview here.
FAST GOSSIP
And finally...
* Justin Bieber and Selena Gomez reunite for disastrous dinner date.
* Bobby Brown pleads not guilty to DUI.
* Kate Moss thinks one of her tattoos is worth a mint.
* Nicollette Sheridan hits legal setback in Desperate Housewives lawsuit.
* Miley Cyrus granted restraining order against home invader.
* The reviews are in for Lindsay Lohan's Elizabeth Taylor TV biopic. Look away now, Li-Lo.
* Watch: Sandra Bullock and Melissa McCarthy play buddy cops in The Heat trailer.
* See Jared Leto in drag for Matthew McConaughey's Dallas Buyers Club.
* Kim Kardashian apologises for Israel, Palestine tweets.
* Britney Spears and will.i.am collaborate on new single Scream and Shout.
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