The former Celtic midfielder has been a steady influence in midfield for the Saints since joining two summers ago, forging an impressive partnership with Morgan Schneiderlin.
The Kenyan is also quoted in the report stating he would be open to working again with Spurs head coach Mauricio Pochettino, who signed him for the Saints.
Wanyama admits he is keen to play in the Champions League, which would give Arsenal an edge over Spurs, or at least counterbalance the Pochettino factor.
The midfielder would have been playing against Spurs this weekend in the Premier League, but is set to miss the game through suspension.
The 23-year-old has a contract at Southampton until 2017, making 28 appearances in the Premier League and scoring three goals, helping them into the seventh position they currently occupy.
At Tottenham, Wanyama's arrival would spell the end for French midfielder Etienne Capoue, who has barely featured in 2015.
Meanwhile at Arsenal he could complement Francis Coquelin, and place added pressure on Mikel Arteta's future, but with Schneiderlin's future also the subject of speculation, Southampton would be desperate not to lose one in case it pushes the other to want to move on, let alone allow the pair to leave.