The Gunners have made contact with Real Madrid about a deal for the World Cup winner but the move has stalled and the Blues have now moved for the 27-year-old.
The Blues have made contact with Real Madrid about discussing a 25 million-euro deal for Khedira, who has only a year remaining on his contract with the Spanish giant.
Chelsea's move comes as Arsenal's pursuit of the Germany World Cup winner has stalled because of his salary demands. Khedira is believed to be pushing for a 250,000 euro-a-week salary plus bonuses.
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That would make him comfortably the highest paid player at Emirates Stadium, where the top earners are the 160,000 euro-a-week duo of Mesut Ozil and Alexis Sanchez, but he would fit in with the Chelsea wage structure for A-list players.
Jose Mourinho had made Juventus star Paul Pogba his top target to partner with Nemanja Matic in central midfield next season, but the Italian club's 75 million-euro asking price has put off the West Londoners.
With new signing Cesc Fabregas penciled in for the playmaker role next season, Chelsea has begun exploring Khedira as a backup option to Pogba given his availability and reasonable transfer fee.
Real is willing to offload the 27-year-old for 25 million euros because his contract expires in 12 months, when he could leave on a free transfer. The Champions League holder has failed to make any headway in agreeing to a new deal with the German and Thursday completed the signing of his countryman Toni Kroos from Bayern Munich.
Chelsea could be encouraged to bid for Pogba following Antonio Conte's shock resignation as Juve manager earlier this week, although the situation for the Serie A champion is fluid because of doubts about the future of his teammate Arturo Vidal. It is unlikely that Juve would cash in on both of its two prized assets.
Arsene Wenger is ready to pursue alternatives to Khedira as he bids to recruit a proven midfield anchorman this summer, with Southampton's Morgan Schneiderlin among those admired by the Arsenal manager.
The Gunners will imminently confirm the signing of Mathieu Debuchy from Newcastle United for 10 million euros plus add-ons and are closing in on a 4 million-euro deal for Nice goalkeeper David Ospina and a season-long loan of Atletico Madrid right back Javi Manquillo.