They claim Antonio Conte has held talks with owner Roman Abramovich and he has made it clear he wants the former Tottenham star at Stamford Bridge.
The Italian boss is concerned that if Eden Hazard joins Madrid this summer they will be left with a huge void in their side.
Hazard is a primary target for Zinedine Zidane’s side and they are likely to test the water with a bid when the transfer window opens.
News of Chelsea’s interest in Bale will cause some concern in Manchester United’s camp though.
Jose Mourinho is also understood to be keen on taking the Wales captain at Old Trafford.
Reports in Spain yesterday suggested Bale had agreed a move to Manchester United, providing they reach the Champions League.
Mourinho’s side are yet to secure Champions League football next season with their best chance of returning to Europe’s elite competition by winning the Europa League.
But with Chelsea on course for the Premier League title and Conte working his magic in west London, Bale could be open to a move to the Blues if they come knocking.
Bale has scored nine goals in all competitions for Madrid this season and is under contract until 2022.
