Vardy was strongly linked with Arsenal who were reported to have triggered the release cause in his contract and offered better personal terms than Leicester, said British media.
"I think Leicester have made a statement and Vardy will stay. We were on the case and interested, but not more than that," Wenger was quoted as saying by beIN Sports on Friday.
"He has decided to stay at Leicester and good luck to him. He came late to the game and in fairness Leicester bought him from Fleetwood and he has been successful. He is 29 and he has chosen to stay at Leicester. You have to respect that."
Growing concerns over midfielder N'Golo Kante's future at the King Power Stadium have intensified after media reports suggested Leicester were close to signing defensive midfielder Nampalys Mendy from French side Nice as a replacement.
With Kante having been impressive for France at Euro 2016, Wenger refused to rule out a move for the combative midfielder.
"It is very difficult to talk about that at the moment. Kante is on the list of many clubs. It is an area where we have many players. We will see what happens in the next two or three weeks," Wenger said.
"It looks like Leicester have bought a player from Nice (Nampalys Mendy) with similar qualities to Kante, so whether they envisage to lose him or not I don't know."
(Reporting by Shravanth Vijayakumar in Bengaluru; editing by Ken Ferris)
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