Poland international Lewandowski has stalled on signing a new deal with Bayern, who are yet to meet this enormous wage demands, which has reportedly allowed Arsenal to initiate talks with the German giants.
Wenger has been keen on the 27-year-old since his days at Borussia Dortmund.
If the Frenchman fails with his pursuit of Lewandowski, it is believed he will shift his attention to compatriot Griezmann, who starred for Les Bleus at Euro2016.
Griezmann, who has a buyout clause of £85.5million, signed a five-year deal with Atletico in June and the Spanish club have no intentions of offloading him anytime soon.
An Emirates source told The Sun: "Arsene is willing to spend on a striker this summer, but only for the right man.
"He has his targets and players he thinks will improve the squad, but this could go down to the wire."