The Mirror claims that the Blues have joined the chase for the full-back after United made a £22m move to bring the 18-year-old to Old Trafford.
Jose Mourinho reportedly sees Shaw as a long-term replacement for Ashley Cole, with the defender fuelling speculation that his days at the club are numbered following a tweet he sent to a fan.
In response to a message on Twitter congratulating Cole on his performance in the 1-0 victory over Stoke City in the FA Cup, the Englishman replied: "Cheers mate. Just don't get used to it."
The newspaper report also suggests that David Moyes is so far winning the race to sign Shaw, having offered the defender the opportunity of immediate first-team action.