According to Sport, Khedira will sign a four-year deal at the Emirates Stadium despite the emergence of Chelsea’s interest.
Contract talks between the 27-year-old’s representatives and Real Madrid broke down and they are now set to sell rather than let him go for free next summer.
Arsenal have already splashed £30m to bring Chile forward Alexis Sanchez to north London from Barcelona, and are on the verge of signing France’s World Cup right back Mathieu Debuchy from Newcastle.
Chelsea boss Jose Mourinho, who worked with Khedira during his time as Real manager, is keen on the defensive midfielder but looks set to miss out.
Khedira missed the World Cup final on Sunday night after he injured himself in the warm-up.
However, he was part of the celebrations afterwards as Germany beat Argentina 1-0 at the Maracana.