The 16-year-old full-back has yet to make a first-team appearance for Saints, although he was an unused substitute in the FA Cup victory over Coventry earlier this month.
He is, however, seemingly attracting admiring glances from a number of clubs in the Barclays Premier League, with Arsenal, Chelsea and United all credited with interest.
But, according to talkSPORT, the Gunners are the strongest contenders to sign Shaw, with chief scout Steve Rowley twice having been to watch the youngster in action.
Arsenal also retain a good relationship with Southampton, having signed both Theo Walcott and Alex Oxlade-Chamberlain from the south coast club.
Oxlade-Chamberlain joined Arsenal last summer for £12 million, and demonstrated his true potential during the weekend defeat to Manchester United.
The 18-year-old winger provided the assist for Robin van Persie in the 2-1 loss, and Arsene Wenger's decision to substitute the teenager has led many Gunners fans to question the Frenchman's judgement.