The 26-year-old's deal runs out in 2023 and United are keen to reward his upturn in form which saw him star for England at Euro 2020, says the Daily Mail.
The left back will be offered a new deal in the next few months as he is seen as a key part of a project to get Manchester United challenging for the Premier League title which they have now gone eight years without winning.
Shaw featured 47 times at left back under Solskjaer last term in all competitions providing six assists and scoring once.
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