The report claims Tottenham have given up hope of signing Real Madrid full-back Fabio Coentrao, with sporting director Franco Baldini now ready to persuade chairman Daniel Levy to move for Shaw in January instead.
But Manchester United and Chelsea are also keen on the 18-year-old left-back, who has impressed in his first full season in the first-team at Southampton.
United are looking for a long-term replacement for current first-choice Patrice Evra, with Shaw on their radar.
But Chelsea are also looking to the future, with Ashley Cole approaching his 33rd birthday later this month.
The England international, like United's Evra, remains one of the most consistent full-backs in the English top flight, but both Premier League clubs want to snap up Shaw, a talented teenager with the promise to play at the very highest level.
Under contract until the summer of 2018, the report claims Spurs are preparing a €12m offer for the defender in January.
But with Cole out of contract at the end of the season, and reportedly a target for MLS side New York Red Bulls, it remains to be seen whether Chelsea or even United lodge rival bids.