The north London club bought in seven players during the summer transfer window at a cost of more than £100 million, but failed to land one of their primary targets, a new left-back.
A move for Real Madrid's Fabio Coentrao failed to materialise, while Benoit Assou-Ekotto departed on loan meaning Danny Rose became first choice.
According to the Haringey Independent, Shaw has been identified as the man to fill the gap and Spurs are prepared to offer £15 million for his services.
The paper adds that Spurs have prioritised a move for the Southampton ace because Coentrao's wage demands have made a January swoop for the Portuguese international impossible.
Shaw is said to be on the radar of a host of leading Premier League clubs, including the likes of Chelsea and Manchester United. Despite his long list of suitors, Shaw penned a new five-year deal with Southampton over the summer.
The Saints are likely to resist any attempts to prise him from their grasp, although they may be tempted by Spurs' January proposal of £15 million while allowing him to stay at St. Mary's until the summer.
The 18-year-old is the latest in a long line of young prospects to have emerged from Southampton's youth academy. Were he to make the move to White Hart Lane, Shaw would follow in the footsteps of Gareth Bale, another Saints graduate who joined Tottenham before leaving for Real Madrid for a world record transfer fee this summer.