Arsenal attempted to lure the 29-year-old to the Emirates Stadium in June after triggering a release clause in his contract with a 20 million pound bid.
However, Vardy, who was away on international duty with England at Euro 2016 at the time, turned down the chance to join the Gunners and instead committed his future to the EPL champions by signing a new four-year deal.
"It's not been put out there that I have always said the boys, the close-knit brotherhood, is unbelievable and for me, there's a lot of unfinished business here," Vardy was quoted as saying by Sky Sports on Saturday.
"I feel that the club is only going one way and that is forward, that's why I want to be a part of it. It wasn't that big the decision when I think about it. It was quite easy and this is where I want to be."
Recent reports in the national press have linked Vardy's Leicester team-mate Mahrez with his own move to Arsenal.
Leicester manager Claudio Ranieri has warned the Algeria winger, who was named the 2015/16 PFA Player of the Year, he risks ending up on the bench if he decides to leave this summer.
Vardy struck a superb partnership with Mahrez last season as Leicester claimed their first top-flight title and he has urged the 25-year-old to follow his lead by remaining at the King Power Stadium.
"I think everyone in the squad would want him here, playing with us, but that's nothing to do with us and it's down to Riyad and the club to sort out," Vardy signed off.
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