The pair enjoyed a night together after he returned from One Direction's Australian. It is the singer's first potential serious relationship since splitting from Caroline Flack, the Sunday Mirror reports.
"They're perfect for each other, they really are. They've been getting to know each other and keeping things quiet but they're close," a source said. "Harry has always had a crush on her and hoped he could get it together with her and they eventually did.
"Everyone hopes they will become an item, but they're both being realistic. Harry's about to start a world tour and Emily's working non-stop."
One Direction have been finishing their second album in Stockholm for the past few weeks and fly to the US this week for their three-month, 62-date tour.
"Harry is spending so much time out of the country he finds it hard to settle down and make a long-term relationship work," the source added. "But he really likes Emily. She's thinking about going to see Harry during the tour and they'll be speaking to each other all the time."
In 2010, the 22-year-old Dancing On Ice contestant, who was recently voted the tenth sexiest woman in the world by FHM, tweeted: "Does Harry from One Direction HAVE to be 16?! Let's pretend he's 18 at least! Then there would only be One Direction he would be going... to the bedroom!"
Styles has been linked to model Emma Ostilly, 18, and Irish photographer Sarah-Louise Colivet, 24, in recent months.
Watch a clip of One Direction on iCarly below: