The highly rated left back is the subject of interest from a number of top Premier League clubs, including Manchester United, but he will not be allowed to leave cheaply.
The Blues made an approach for the left back earlier in January and were told Shaw would not be sold during the winter transfer window under any circumstances.
However, the departure of Southampton chief executive Nicola Cortese on Wednesday has cast Shaw’s immediate future into doubt.
Chelsea, which was preparing a big-money summer swoop, could look to take advantage of the uncertainty at the south coast club and move to land the England Under-21 international as a long-term replacement for club legend Ashley Cole, who shares Shaw’s representatives and is out of contract in the summer.
Jose Mourinho, the Blues manager, is a huge fan of Shaw - a lifelong Chelsea supporter - and the Londoners are unlikely to be put off by Southampton’s 25-million-pound valuation.
Manchester United has also asked about Shaw while Tottenham have registered its interest in the 18-year-old and was at one stage thought to be the favorite to land the Saints academy product.
United boss David Moyes is likely to join any bidding war for Shaw as he also hunts for a new left back to replace Patrice Evra, who is also out of contract at the end of the season.
The Premier League champions have looked at deals for Everton star Leighton Baines and Real Madrid's Fabio Coentrao, but in both cases have so far been deterred by the prohibitive demands of the players' clubs.
Shaw’s priority is guaranteed first-team football at a Champions League club and while he is happy at Southampton, he could look to accelerate his plans to leave the club having previously ruled out a winter move.
Cortese’s exit has raised doubts over the future of manager Mauricio Pochettino, despite the Argentine’s insistence on Thursday that he will remain in charge, as well as the future of key players including Shaw and Adam Lallana.
Shaw signed a new five-year contract with Southampton after turning 18 last July and has stated he is in no rush to move clubs, with Pochettino’s side currently ninth in the Premier League table and eyeing a place in Europe next season.
He played 28 times for Southampton last season and has made 19 appearances so far this campaign in all competitions.