Vardy is regarded as a club legend at the King Power Stadium after scoring 171 goals in 427 appearances for the Foxes.
The former England international played a starring role in Leicester's Premier League triumph in the 2015-16 campaign.
In addition to lifting the Premier League trophy, Vardy has also won the Community Shield and FA Cup during his 11-year stay with the club.
Vardy struggled to find his best form last season, though, mustering just three goals in 37 top-flight appearances.
Leicester were ultimately relegated to the Championship, bringing an end to their nine-year stay in the top flight.
With 12 months left to run on his contract and question marks surrounding his future at the King Power Stadium, Vardy has been presented with the chance to join Saudi Pro League club Khaleej FC.
However, according to Sky Sports News, the experienced forward has turned down the offer because he does feel that the move would suit his family.
Vardy is yet to meet with new Leicester boss Enzo Maresca, but the Italian has made it clear that he wants the striker to stay at the club.
"He'll be staying with us yes," Maresca told The Telegraph. "That's what we're currently doing, looking at the squad and planning for next season."
While Leicester are set to retain the services of their club legend, they are also working to bring in new additions.
Conor Coady is reportedly close to signing from Wolverhampton Wanderers for a £7.5m fee, which could rise to £8.5m in add-ons.
There are also on the verge of recruiting USA goalkeeper Zack Steffen on a permanent deal from Manchester City.