The former Spice Girl and single mom, 40, bonded backstage with Brand at the Closing Ceremony for the London Olympics -- and the twosome were subsequently photographed out together at Hampton Court Palace with Bluebell, the singer's 6-year-old daughter from a previous relationship.
"Russell and Geri are just friends and have been out a few times of late," one insider tells Us Weekly. "A few media are prematurely reporting that they are in a serious relationship, [but it's] a little early for all that type of speculation."
Adds a second source: "They're testing the waters. It's very early days."
Halliwell herself recently played coy about a new relationship in an interview with British mag Hello! "I'm not going into details," she said, giggling. "Right now there's nothing serious. I was a late starter on the romantic front. I didn't start dating until I was in my 30s . . . I quite like having somebody to cuddle. I like that intimacy, it would be wonderful."
For his part, the Brand X star certainly is familiar with flamboyant pop singers: He and Katy Perry finalized their divorce last month. He filed to end their marriage in late December after just 14 months of marriage.