The left-back was replaced by Kieran Gibbs as a precaution midway through the second half of England's 3-1 friendly victory at Celtic Park.
Shaw was troubled by a hamstring injury earlier this season following his move from Southampton to Old Trafford, and the problem flared up again on Tuesday.
The 19-year-old told Sky Sports: "I picked up a few knocks during the game and my left hamstring, which I hurt at the start of the season, I could feel a little bit.
"We had a discussion at half-time over whether to bring me off then but it was a game I didn’t want to come off in so I played the extra 15 minutes and then Roy [England manager Hodgson] decided to sub me off.”
When asked if he would be in contention to face Arsene Wenger's side in the Premier League clash at the Emirates Stadium, he added: "We will just have to see, but I am hoping to be fit."
Danny Welbeck also suffered a hamstring injury in the same game and it remains to be seen if the Arsenal striker will be ready to play against his former club.
United winger Angel Di Maria is expected to be fit after playing down the extent of a knock he picked up during Argentina's defeat to Portugal at Old Trafford on Tuesday.
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