The 17-year-old - dubbed the 'next Gareth Bale' - is the latest talent of the club's production line, and has started Saints' last three games after making his Premier League debut as a substitute against West Brom earlier this month.
Club captain Adam Lallana, another of the side's homegrown players, hailed Shaw saying: "There's big similarities between him and Bale. He has got to keep up his fitness and he is going to go far."
Nigel Adkins is keen to reward the exciting left-back for making his first-team breakthrough this season, and keep the Premier League's big guns at bay.
Gunners boss Arsene Wenger has been credited with an interest in signing Shaw, who could follow in the footsteps of former St Mary's scholars Theo Walcott and Alex Oxlade-Chamberlain by moving to the Emirates Stadium.
However, Arsenal will face stiff competition, with Manchester United also keen on recruiting the talented teenager, with Sir Alex Ferguson known to be in the market for a long-term replacement for Patrice Evra at Old Trafford.
Big-spending champions Manchester City are also believed to be in the mix, with Roberto Mancini eager to bring Europe's brightest up-and-coming talent to the Etihad Stadium, while Chelsea have had offers for Shaw turned down in the past, under the former Stamford Bridge stewardship of Roberto Di Matteo.
Southampton recently handed a fresh five-year contract to promising midfielder James Ward-Prowse, and Shaw - who only turned 17 in July, and is currently the youngest player in the Premier League - is in line to receive similar terms.