The 29-year-old England international has not made a final decision, but is understood to favour extending his time at the King Power Stadium.
Arsenal made a reported bid of £20m for Vardy, activating a release clause in his contract and offering him a deal reported to be worth £120,000 a week.
Leicester responded with an offer said to be worth £100,000.
Arsenal do not plan to improve their offer and are thought to be pursuing alternative options.
Vardy, who joined Leicester from non-league Fleetwood Town for £1m in 2012, signed a three-year deal in February, and the Foxes' new offer would extend that into a fourth year.
Arsenal's offer was a three-year contract with the option of a fourth.
Vardy, whose tally of 24 league goals last season was bettered only by Tottenham's Harry Kane, is currently with the England squad at Euro 2016.
The Gunners asked Vardy to give them an answer before he flew to France, but he wanted to focus on the competition.
Leicester striker Jamie Vardy is set to reject an offer to join Arsenal and stay with the Premier League champions.