The Blues are trying to find a long term replacement for England left back Ashely Cole as they are not keen on making Ryan Bertrand or Cesar Azpilicueta their main man in that position.
Chelsea have been in contact with Southampton and Shaw over a move but it will take big money to draw the south coast club into a deal.
Shaw however has told the West London club that a move may be viable if he is guaranteed first team football, leaving the future of injured Cole in tatters.
The highly rated wing back is being watched by numerous clubs not just in England but across Europe and has been tipped to go to the World Cup next summer over either Cole or Leighton Baines.
Chelsea will have to make their move sooner rather than later to prevent other clubs such as Bayern Munich, Manchester City and Real Madrid stepping in, however it will take big money to lure Southampton into a deal have strapped their youngster down to a long term contract.