The Gunners chief made an £18.5m approach for the 24-year-old over the summer, but saw his move thwarted by the Bundesliga giants who were unwilling to sell.
But, after failing to secure an alternative defensive midfielder Wenger is ready to up his offer for the German international to £25m.
The Londoners have already swooped for Mesut Ozil, but are keen to bolster their midfield ranks further once the transfer window opens.
Wenger is a big fan of the classy enforcer, and sees him as the ideal man to compliment the more creative players he already has in his engine room.
Bender may be tempted by a move to the Premier League, having spent four years at the BayArena.
Arsenal are also keen on a number of other players based in Germany.
Borussia Dortmund trio Ilkay Gundogan, Lukasz Piszczek and Marco Reus have all caught Wenger’s eye, and the Frenchman is believed to be lining up potential moves ahead of the turn of the year.
Gundogan is considered to be his top target, while Piszczek is seen as an ideal replacement for Bacary Sagna.
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