The 30-year-old was stretched off the pitch after he fell awkwardly during United's 1-0 win over Everton at Goodison Park on Saturday.
Silvestre is the third United player to tear cruciate knee ligaments this season, following similar injuries hitting Park Ji-Sung and Ben Foster.
The blow will see manager Alex Ferguson have limited options on the left flank, which leaves Patrice Evra as the only natural left-back after Gabriel Heinze's departure this summer.
However, the Red Devils confirmed that injured duo Wayne Rooney and Owen Hargreaves will be fit for Wednesday's Champions League opener at Sporting Lisbon.
"They'll both be ok for Wednesday," Sir Alex told MUTV.
United welcome back Rooney, who has been missing since the opening day of the new season, after the England striker broke his metatarsal against Reading.
"We took medical advice on Wayne and as he had only started training on Monday, we felt that five days' training was maybe a bit short," added Ferguson.
"But he trained on Saturday morning, Sunday and he'll train right through until Wednesday. By that point he will definitely be available for the match against Sporting Lisbon."