It is understood that the Gunners swooped in to try and sign Vardy after Manchester United did not show any intent of wanting to acquire his services. Red Devils manager Jose Mourinho seems to have his sights set on only one striker, Zlatan Ibrahimovic.
With Vardy having a £20m release clause in the new four-year deal he signed in February, Arsenal are willing to meet that clause in order to add the 29-year-old to their roster.
Vardy, who is currently on duty with England ahead of Euro 2016, enjoyed a lot of success in the 2015/16 season as he scored 24 goals and notched up eight assists in all competitions.
When asked about his future last month, Vardy told Sky Sports: "I just won the Premier League and now I'm in the Champions League. I'm happy here."
While it will be seen whether Arsenal and Leicester City will meet, the Gunners have already signed VfL Borussia Mönchengladbach midfielder Granit Xhaka and will be looking to continue bolstering their squad after finishing second in the Premier League.