Jonas and Greene were spotted on a Disneyland date and holding hands at a Walgreens, the source said.
He "is head over heels for (Ashley)," the source told the magazine. "He thinks she is down-to-earth and absolutely beautiful. He hasn't been this into a girl in a long time."
The two are getting more serious: The source tells People Greene, 23, took Jonas, 21, home with her to meet the parents in Florida.
Jonas's rep had no comment, the magazine said.
'Maze Runner' brings in $11.3M Friday
Dakota Fanning says her mother is no stage mom