Old Trafford chiefs need to make room on their wage bill – currently the largest in the Premier League - ahead of adding more new faces at the end of the campaign.
Shaw signed a new deal at United in October 2018 but his inability to stay fit continues to be a major bone of contention for a player who earns £150,000-per-week.
The Red Devils are tracking Leicester City and England left-back Ben Chilwell, who they know would set them back at least £50m to entice the Foxes into a sale.
Shaw cost United £30m when he was signed from Southampton back in 2014, suffering a devastating broken leg soon after against PSV Eindhoven in a Champions League game.
The 24-year-old battled back but has been plagued by repeated injuries, struggling to recapture his best form and being publicly berated during Jose Mourinho's time in the United managerial chair.
Shaw's present agreement runs until 2023 and there are some at United who feel it might be the best time to cash in on the defender.
The Manchester giants know annual revenues will fall below £600m if boss Ole Gunnar Solskjaer fails to secure Champions League football this term.
There are also concerns they will still struggle to offload striker Alexis Sanchez - without handing him a multi-million pound severance package.
The Chilean flop earns a mammoth £505,000-per-week, with United currently picking up more than half of his salary while he is on a season-long loan at Inter Milan.
The Italian giants cannot afford to keep the former Arsenal man unless his parent club are prepared to accept a massive financial hit, which is why fringe players such as Shaw are seen as a way of raising future transfer funds.