The Serbian defender could also miss a potential semi-final first leg with either Barcelona or Schalke should United qualify.
But he would hope to be fit for a possible crunch Premier League clash with Chelsea at Stamford Bridge on 26 April, and the second leg of a potential Champions League semi-final on 29/30 April.
It had been feared the Serbian defender might not play again this season but United have confirmed Vidic should return within three weeks.
"There is no major damage to the left knee, and he will be out for two to three weeks," said a United spokesman.
Vidic has formed a superb partnership with Rio Ferdinand at the heart of defence and has been a key part of United's chase for a Premier League and Champions League double.
Manager Sir Alex Ferguson has plenty of defensive options available to him with O'Shea and Gerard Pique both having played at centre-back in recent weeks, while switching Wes Brown from right-back is another alternative.
Mikael Silvestre returned to the first-team squad in Rome following a knee injury while Gary Neville is also close to a return.
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