Vardy, who is in England's 23-man squad for the Euro 2016 finals in France that start next week, scored 24 Premier League goals last season as Leicester won the top-flight crown for the first time in their history.
The 29-year-old was chosen as Player of the Year by the Football Writers' Association and was named in the Professional Footballers' Association's team of the year.
Former England forward Gary Lineker, a Leicester fan, has urged Vardy not to leave.
"Stay with the Champions @vardy7 don't join the perennial also-rans," Lineker said on his Twitter account.
Arsenal have not won the league since 2004 and finished second this season, 10 points behind Leicester.