According to The Metro, Bayern Munich are now serious rivals to Arsenal for the 27-year-old midfielder, who is expected to leave the Bernabeu this summer following the arrival of his Germany international teammate Toni Kroos.
But Real Madrid, who accepted a £24m offer from the Gunners earlier this month, want Khedira to choose Arsenal ahead of Bayern Munich.
Bayern, now managed by former-Barcelona boss Pep Guardiola, are not going to bid for the Germany international but intend to wait for his contract to expire at the end of next season and sign him on a free transfer.
Bayern believe if they make it clear to Khedira that they will sign him next summer, he'll sit tight at Real Madrid this season and pen a contract to play for them from the 2015/16 season onwards.
Real Madrid are obviously keen to avoid this situation, as they would lose a key first-team player for nothing. As a result, the Metro speculates that they may be willing to do a deal with Arsenal, lowering the asking price to compensate for Khedira's large wage demands.
An initial move from the Gunners hit a stumbling block when it came to Khedira's demands - he wanted £150,000-a-week at first. He has since made it clear he would accept £134,000-a-week but that's still above Arsenal's budget and they have yet to decide whether to offer him that package.
Chelsea have also been linked with Germany star, who won the World Cup with his country in Brazil this summer. Khedira was first signed by Real Madrid by then-Madrid manager Jose Mourinho.
Defensive midfield options
Arsenal boss Arsene Wenger is looking to sign a defensive midfielder before the close of the summer transfer window, and Khedira is one of several options. He is the most high-profile, but is also the most expensive.
Alternatives include Southampton's Morgan Schneiderlin, eager to leave the Saints after his Twitter outburst yesterday, but the Frenchman wants to link-up with Mauricio Pochettino at Tottenham.
Bayer Leverkusen's Lars Bender is another potential target for the Gunners.