The German playmaker was a target for Manchester United until recently, but once it became apparent to the Red Devils that they may be unable to lure him across to Old Trafford, their interest cooled.
Since then Real have moved into pole position, and are thought to be close to snapping up the Germany international – who has just 12 months to run on his contract.
But now it is understood that Chelsea have weighed in with a late offer in a bid to steal him away.
Jose Mourinho’s proposal matches that of Los Blancos – his former club – and it may now be down to the player to make the decision.
Kroos could net a £125,000-per-week salary at Chelsea.
The move would come as somewhat of a surprise, with the Blues having already snapped up Cesc Fabregas this summer.