Rashford was substituted in the 80th minute of Saturday’s 2-0 Premier League win over Everton at Old Trafford and immediately headed to the dressing room for treatment.
United say the England international “will be unavailable for a few games, but is expected to be back for the season run-in”.
Rashford has been a key part of the club’s revival this season under Erik ten Hag by scoring 28 goals in all competitions, including one at Wembley in February which helped them lift the Carabao Cup, the first available trophy of the season.
“Some things you can’t avoid, but this was avoidable,” Ten Hag said. “Why has the Premier League given us the late Sunday night game and the early Saturday game? I think it’s not right.
“You are running a risk; players can’t recover that quickly. All science and research will tell you players need a certain time to recover.
“It is also part of the schedule that we are now finding ourselves in this situation and now we can only pray he [Rashford] is not dropping off.”
Asked whether he was guilty of overplaying Rashford, and whether that was indicative of a lack of depth in the squad, Ten Hag said: “Not any manager in this stage of the season will rotate so much.
“I know that is why this is the toughest league and in all the leagues and competitions we were in then you need a squad to rotate to a certain point, but this was avoidable, it was not necessary to set the schedule like we have now.
“Other things are more important from a sporting element like protecting the players. Today was saw an entertaining afternoon but players can’t do that when they are not fresh.
“We created a lot of chances but missing chances is due to a lack of freshness in the final moment. You risk injuries so we have to protect the players, and that is also in the interest of what the audience want to see. You have to have the best players on the pitch.”