The source said the new couple had taken a trip together to Palm Springs, California. Not coincidentally, both Kourtney and Travis have shared images of what looks like the backyard of her mother, Kris Jenner's, Palm Springs abode.
"They've been friends for a long time, but it's turned romantic," the source continued.
Sharp-eyed "Keeping Up with the Kardashians" fans will recall Travis has appeared on the show.
So far, neither has confirmed or denied the story.
Kourtney and her ex, Scott Disick, are the parents of son Mason, 11, daughter Penelope, 8, and son Reign, 6. Barker has a stepdaughter, Atiana, 21, a son, Landon, 17, and a daughter, Alabama, 16, with his ex-wife, Shanna Moakler.