This is despite the Gunners signing Egyptian midfielder Mohamed Elneny, who plays in the same position.
The Mirror claim experienced trio Mikel Arteta, Tomas Rosicky and Mathieu Flamini are all likely to be allowed to leave, creating a space in the squad for Wanyama.
It was reported by SkySports last week that Wanyama is considering his future at the Saints but will not make a decision until May, ending speculation he could move on this January during the transfer window.
Southampton have cashed in on their assets in previous summers, with Luke Shaw, Adam Lallana and Morgan Schneiderlin among those to leave the club.
Ex-Celtic star Wanyama would not be the only player to make the move from Southampton to Arsenal, with Theo Walcott, Alex Oxlade-Chamberlain and Calum Chambers signed by Arsene Wenger over the past decade.