These pics were shot on Richard Branson's Necker Island in the British Virgin Islands ... where the 2 of them were part of a tennis fundraiser for the Inspiring Children Foundation. Jewel is the founder of the org, and Kevin was a guest.

Here's what we know. The 2 flew to the Caribbean together and hung out a lot for more than a week. As one source put it, "There was definitely something going on. They were flirty, and when they were together, it was like they both just lit up."

One source added, "You could tell they were trying to be discreet, but anybody who saw them could tell something was going on."

Jewel had mentioned Kevin on social media, talking about the event, writing ... "it's an incredible time, and one I use to relax, rest and play w my son! @kevincostnermodernwest was kind enough to mentor our kids this year."

They played doubles with Sir Richard, who, as she says, "plays a mean game!"

We're told the 2 also flew back together.

As we reported, Kevin, 68, recently settled his bitter divorce from Christine after 18 years of marriage. Jewel, 49, split from her husband Ty Murray, back in 2014. One source told us Jewel is "picky," adding, "She wants a good man, and Kevin fits the bill."

Our sources say the relationship is "rather new," but everything we're hearing is that they are definitely a thing.