The England international was forced off after 34 minutes of the Premier League clash, and "scans have shown the injury is more serious than first thought," according to TalkSport.
Ashley Young replaced Shaw on Saturday and is the most likely player to deputise for the 24-year-old.
Manager Ole Gunnar Solskjaer played down concerns over Shaw's injury after the match. He told reporters that he was hopeful the full-back would return for their next name against Southampton.
If the full-back is out for a month, he will miss games against former club Southampton, Leicester City and West Ham United.
The injury would also rule him out of international action. England manager Gareth Southgate is due to announce his squad on Thursday for the Three Lions' UEFA 2020 European Championship qualifiers against Bulgaria and Kosovo.
Rob Dawson at ESPN FC reported that Shaw "will undergo more tests this week" amid fears "he could be sidelined for closer to six weeks."
The news will be a further injury blow for Solskjaer. The Norwegian is already without full-back Diogo Dalot, who is not due to return until after the international break:
Diogo Dalot has picked up an injury in training and won't be available until after the international break. "He'll be out for a little while, he's had the problem for a couple of weeks." Everyone else is fine apart from long-term absentees Eric Bailly and Fosu-Mensah. #MUFC
Meanwhile, France forward Anthony Martial also picked up a thigh injury against Crystal Palace, and United are "waiting on the results of scans" to determine the severity of the problem, per Jamie Jackson at the Guardian.
Manchester United will be aiming to get back on track on Saturday after a demoralising stoppage-time defeat to Palace. The loss was the first time they have been beaten by the Eagles in the Premier League:
The result leaves the Red Devils with just four points from first opening three games and without a win since thrashing Chelsea 4-0 on the opening weekend.