Taylor Swift and Travis Kelce Hung Out Several Times in a 'Very Private Setting'

Taylor Swift and Travis Kelce are not "officially" dating, at least not yet ... but they are definitely headed in that direction, and the early returns seem promising.

Sources with direct knowledge tell TMZ ... Taylor's appearance at Travis' NFL game Sunday was NOT the first time they were together. We're told they have been together "several times" over the last few months.

While Taylor's show of support was very public, our sources say their person-to-person interactions before Sunday were always in a "very private setting."

Still, it's interesting Taylor is already meeting Travis' parents ... our sources say Sunday marked Taylor's first introduction to TK's mother, Donna, and father, Ed.

As you know, TayTay watched Travis score a touchdown from a suite with his family ... and cameras caught their excitement.

We're told Taylor's time with Donna, Travis' mom, went "very well" ... and that the matriarch calls Taylor "a lovely person."

Taylor has yet to meet Travis' older brother Jason -- sounds like that will have to wait -- he's busy playing for the NFL's Philadelphia Eagles.

Taylor and Travis smiled for the cameras as they left Arrowhead Stadium together after his Kansas City Chiefs pummeled the Chicago Bears, 41-10 ... leaving in a convertible. We're told they took a shot with some friends before leaving the stadium.

We didn't see any PDA from Taylor and Travis in public, but our sources say they were holding hands at the restaurant he rented out for his family, friends and teammates after the game. At one point Taylor was sitting on Travis' lap.

Taylor fit in well with Travis' friends and family ... she came to the game alone -- bodyguards excepted -- and was chatting up everyone.

Taylor's upcoming tour dates could make a full-blown relationship tricky ... she's about to go back on tour in a few weeks, and then in November she heads out for her tour's international leg. Meanwhile, the NFL will consume Travis' life, potentially through February's Super Bowl.