The former England international is keen on a reunion with his former Leicester manager Brendan Rodgers at Parkhead.
Vardy remains a free agent after coming off contract at the Foxes over the summer.
A source told The Sun: “Jamie still believes he can play week in week out and has a lot to offer. He’s a model professional and is keeping himself in great condition.
“He’d love to move to Celtic and really has his heart set on that move.
“He loved playing under Brendan and thinks he could be a massive benefit to the team up there. In fact, ever since it became clear Celtic were up for the deal Jamie’s been desperate to make it happen.
“But other clubs are keen to so time will tell. But one thing is for sure he won’t be without a team at the end of August.”
Indeed, the Hoops aren't the only top-level option emerging for Vardy, 38, with Brentford and West Ham also keen.