The online bookmaker now make the Gunners just 1/7 to secure his services this summer, with reports in Spain suggesting they have beaten off competition from Chelsea, who have drifted out to 8/1.
Khedira, 27, is thought to have already agreed a four-year deal, with a fee of around £23million potentially taking Arsene Wenger’s summer spending towards £60m following the acquisition of Barcelona and Chile forward Alexis Sanchez.
Khedira has just one year left on his Real Madrid contract.
Sky Sports' Spanish football expert Guillem Balague says the Champions League winners want him to sign a new deal but Khedira's agent says he is in no rush to negotiate.
Seemingly convinced the Khedira deal is imminent, Sky Bet have also responded by cutting Arsenal’s Premier League title odds from 11/1 to 13/2, leapfrogging Liverpool in the market.
Sky Bet’s Chris Spicer said: “Such has been the conviction of the betting on a deal being done, we’re assuming it is more than just speculation and have therefore trimmed Arsenal’s title odds again, having previously also done so when the Sanchez deal was confirmed.
“The addition of Khedira could arguably be even more vital given his physical presence in midfield and much-need experience, having won the World Cup and Champions League in a matter of months.”