The Portugal international sustained the problem in Sunday's Community Shield defeat to Chelsea at Wembley.
Nani had put United in front, but he then injured himself in a challenge with Chelsea captain John Terry later in the match.
Reds Devils boss Sir Alex Ferguson fears the 22-year-old could be looking at a prolonged period on the sidelines.
Ferguson said: "He's out for a while, the worry being that we couldn't put it back into place."
Hargreaves concernsNani had scored United's opening goal, but the Premier League champions were held to a 2-2 draw and were then beaten 4-1 in the penalty shoot-out.
Meanwhile, United have dismissed speculation Owen Hargreaves has suffered a setback in his recovery from operations to both knees.
It was suggested Hargreaves was flying to Germany for specialist treatment in Munich and his career was once again in jeopardy.
However, the reports have come as news to Ferguson, who has privately confirmed the player is still in the United States continuing his rehabilitation programme.
The 28-year-old has not played since last September, when he began looking for a cure to the tendinitis problem he has suffered from since before his £18million switch from Bayern Munich in 2007.