The 22-year-old has struggled to make an impact at Old Trafford since his £27m move to Manchester United from Southampton in the 2015 summer transfer window.
Shaw made some progress in his second season under Louis van Gaal before a horrific leg break in a 2-1 loss to PSV Eindhoven in the Champions League group stage in September 2015.
Manchester United appointed Jose Mourinho as Van Gaal’s successor in May 2016 ahead of the England international’s return to full fitness from the leg break.
Mourinho hasn’t been afraid to criticise Shaw since taking over the reins at Old Trafford over the past 18 months amid the defender’s fitness struggles.
The Manchester United manager was critical of the left-back after Shaw was withdrawn at half-time of last weekend’s 2-0 victory over Brighton and Hove Albion in the FA Cup.
Former Chelsea FC defender Le Saux believes Shaw should consider a move to Stamford Bridge at the end of the season.
“I do believe he has a very bright future and a lot of top level clubs will be looking at him right now,” Le Saux told Betfair, as quoted by Metro.
“If anyone can get him back to the level he was at when playing for Southampton then he would be a brilliant buy and I don’t think it would take too long for him to get back into the England squad either.
“I think he’s definitely a player who should be playing for one of the top six clubs and any one of those will be sniffing around I’m sure.
“He’s a top-level player who has had setbacks, but he just needs to head to a club that will foster his talent. Chelsea could be one such team of course.
“It depends what system they play and they’ve got players there now that he would have to get past, however I do think at full fitness Chelsea would be interested as he’s a dynamic, fast player who can cover the pitch.
“As I say though, I think anyone of the big clubs could be his future.”
Shaw is a self confessed Chelsea FC supporter, although he opted to move to Manchester United ahead of interest from the defending Premier League champions in the 2015 summer transfer window.
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