The young left-back has been linked with a host of leading Premier League clubs after establishing himself as a mainstay of the Southampton first-team.
Both Chelsea and Manchester United have been linked with the 18-year-old, who penned a new long-term contract with the Saints over the summer.
According to Goal it is Tottenham who are the most likely to land Shaw - although only in the summer after Southampton ruled out selling in January.
The website claims Tottenham have been putting in the groundwork with Shaw's family and representatives in order to pave the way for a move to White Hart Lane.
Either team, or indeed Manchester United who are in the market for a new left-back, would have to pay in excess of £10 million to land their man.
Chelsea are believed to see the Southampton man as a long-term solution for Ashley Cole, with doubts still hanging over Ryan Bertrand's ability to make it at the top level.
Spurs are also likely to bring in a new left-back sooner rather than later with Danny Rose currently their only senior specialist in that position.
Manchester United are expected to go back in for Leighton Baines in the winter window having failed to complete a deal for the Everton man in the summer. They also failed to secure the services of Real Madrid's Fabio Coentrao - who Tottenham are also interested in - on deadline day.
Shaw has previously insisted he is no rush to leave Southampton and follow the likes of Gareth Bale, Alex Oxlade-Chamberlain and Theo Walcott away from the south coast club to a major Premier League team just yet.