However, it was only in February that Vardy signed an improved three-and-a-half-year contract with Thai-owned Leicester.
The Guardian said Leicester could offer Vardy a new deal if they think that is what it will take to keep him at the Midlands club.
It seems the forward now has a momentous decision to make. A well-liked member of the Leicester squad - the players celebrated their title-winning success with a party at Vardy's house.
However, he only became a full-fledged professional player in 2012, after nine years in non-league football, and if Arsenal are offering significantly more money than he's getting at Leicester, Vardy could find it hard to turn down the Gunners.
He said last month amid transfer speculation: "We've just won the league and will be playing in the Champions League next year. I am happy here."
The Guardian also reported that Manchester United had shown interest in Vardy but remained committed to Zlatan Ibrahimovic as its No.1 target. Sky Sports said United had in fact passed up the chance to sign Vardy, as new manager Jose Mourinho only wants to sign one striker during the transfer window.
Vardy, signed for £1 million from non-league Fleetwood in 2012, was named the Football Writers' Association Footballer of the Year for 2015-16. He is now an England international with three goals from eight games and is Euro 2016 bound.
Arsenal have already signed Swiss midfielder Granit Xhaka from Borussia Monchengladbach.
Leicester have also faced prolonged transfer speculation over N'Golo Kante (Arsenal) and Riyad Mahrez (Barcelona).
