The 23-year-old has reportedly clashed with United boss Jose Mourinho on multiple occasions, and it now appears that the left back, who has been previously criticised by Mourinho for his lack of fitness and poor attitude, has been made available for transfer in the summer window.
The Times report that the Toffees were interested in his services, but Shaw turned down the possibility of moving to Merseyside.
Shaw was once thought of as a future England star when he made the move to Manchester from Southampton for £30m in 2014 (becoming the most expensive full back in the world at the time), but a horrific challenge from Hector Moreno during a Champions League match in 2015 broke the defender's leg in two places, and he's never quite fully recovered.
In his four seasons at Old Trafford, Shaw has started just 37 league games. In total, Shaw has played 43 matches for the Red Devils in the Premier League, registering three assists.
As such, Mourinho is looking to move the Englishman on this summer, but Shaw has one season left on his £5m-a-year contract and reportedly intends to leave as a free agent in 2019 - thus his decision to reject a move to Everton.
This complicates matters for United, since they'd ideally like to move Shaw to free up space and funds for Juventus' Alex Sandro - though the club is so far refusing to meet Juve's asking price for the Brazilian.
From Everton and new manager Marco Silva's point of view, a move for Shaw is in line with their attempts to instil some young blood in an ageing squad - but it remains to be seen if the club are willing to wait for Shaw to become a free agent.
Instead, the Times report that the Toffees have opened negations with Barcelona over the transfer of 24-year-old France international Lucas Digne, who has been made available by the La Liga champions this summer.