Chelsea were keen on the German World Cup winner throughout the summer transfer window and while no official bid was made then, The Blues have maintained their intereset and are believed to have now made a formal offer to lure Khedira to Stamford Bridge in January.
Khedira will be out of contract at the end of the year and Real Madrid are desperate to sell the 27-year-old during the upcoming transfer window in the new year to prevent him from departing for free next summer.
Arsenal meanwhile have rekindled their interest in the last few days but Chelsea are believed to be leading the race.
By Ollie Carroll
Twitter: @Ollie_Carroll_