EAST RUTHERFORD, N.J. — You have probably heard the Giants have a new running back. You may already be one of his 975,000 followers on Instagram. Perhaps you know he can squat 650 pounds and hurdle 6-foot defenders. Maybe you saw him — a lot of him — in ESPN the Magazine’s Body Issue.

His name, Saquon Barkley, has become more than a preseason buzzword. It is bordering on an infatuation.

And so the N.F.L.’s annual hype-a-thon — also known as training camp — has been kicked into overdrive at Quest Diagnostics Training Center, where the Giants are struggling to temper expectations for a rookie out of Penn State who is already reaching mythic status from a fan base eager to forget last year’s 3-13 season.

After all, in June, Barkley’s No. 26 was the best-selling jersey in the N.F.L. This was before one of the team’s owners, John Mara, called him the most glorified rookie the franchise has had since Lawrence Taylor. But it was after Dave Gettleman, the Giants’ general manager, said that Barkley’s skill set was as though he had been “touched by the hand of God.”