Having played 52 minutes in United's season opener on Sunday, Vidic was forced to be replaced by Jonny Evans.
United manager Sir Alex Ferguson, said
: "It is difficult to say with calves but it is longer than we thought."
Fellow central defender Rio Ferdinand also picked up a hamstring injury during United's 2-1 victory on the weekend, and was replaced by United newcomer Phil Jones. Ferdinand, however, may return by the end of August.
: "Rio is not as bad as we thought and he could be ready for our game with Arsenal. I don't think he'll be ready for Monday, but it's amazing how assessments can change in the space of a few days."
Although Chris Smalling and Jones are a combination for the future, Ferguson will at least require the experience of Jonny Evans while Vidic and Ferdinand recover.
Jones and Evans are expected to start in central defence for Monday's clash with Tottenham Hotspur, as Smalling will most likely fill in at right-back for Rafael da Silva, who is still on the injury list with a dislocated shoulder