The Frenchman is concerned by an imbalance in Arsenal's midfield caused by Alex Song's sale to Barcelona and is weighing up a bid for a more physically imposing figure than Mikel Arteta, Jack Wilshere or Aaron Ramsey.
Though Wanyama's performances have seen his asking price rise above £10m, the 21-year-old's wages are comparatively low and he has refused to extend a contract with Celtic that expires in 2015.
Another option for Wenger is Mohamed Diame. Although the Senegal international only signed for West Ham United in June, a £3.5m release clause was written into his new contract and the 25-year-old is open to a move across London.
Wanyama also has the advantage of offering cover at centre-back – another problem area for Wenger. Celtic manager Neil Lennon frequently used the African in defence during the player's debut campaign in Scotland.
Awarded a regular starting berth in midfield this season, Wanyama has scored seven times in 32 appearances.
Manchester United, who are looking to recruit in central midfield in June, have also scouted Wanyama extensively.
One player Wenger has signed up during this transfer window is Theo Walcott, who has expressed his delight at his contract extension.
"I am very pleased that it's sorted. I always said I wanted to stay and I'm happy that we came to an agreement," he said.
"It has been tough, not just for me, but for my family, my partner, they were all getting hit with it every week. I'm pleased it's over."
Walcott now feels Arsenal should show more self-belief. "We need to start realising how good we are," he said. "We showed some great stuff in the second half at Chelsea but it's time we did that from the start. There's a lack of belief as to how good we can be. We've got some very good players. The players don't believe they are good enough at times.
"In the first half we were too slow, sloppy closing down and we've got to show more commitment. As soon as we pressurised them a bit more, pushed up higher (we were better)."
Arsenal play West Ham at home tomorrow and Walcott admits they must start picking up points. "(Playing in the Champions League) is one of the reasons I decided to stay. I still think we will (make the top four).
"The gap doesn't worry us. We can't worry about Spurs. We hope they drop points. But we've got to concentrate on ourselves, win games and we've got a big game on Wednesday when we have to pick ourselves up."