Of course, there's a lot of buzz that the former Penn State running back is being targeted by the Giants -- who have the #2 overall pick in the NFL Draft.

Is it a good sign for NY Giants fans? Sure ... but don't go celebrating just yet.

Sources close to Barkley tell us the sweatshirt was a gift given to him from the Giants when he met with the team -- and it just happened to be laying around when he needed a shirt to go get a haircut.

FYI -- we're told players are often given team swag during pre-draft visits.

In other words ... Barkley's position is that he'd still be happy to be drafted by any team.

But S.B. is widely considered the best RB in the draft -- and ya gotta think Eli Manning and Odell Beckham Jr. are gonna be stoked when they see these pics.