The defender, a £16m summer recruit from Sporting Lisbon, picked up the injury in last week's 1-0 defeat against Manchester City to add to United's problems at the back.
"We have always a risk that it happens again. Conservatively, it can be less than six weeks. If we have to operate, 12 weeks."
Louis van Gaal on Marcos Rojo
Van Gaal is already without the suspended Chris Smalling while Phil Jones (shin) and Jonny Evans (ankle) are both sidelined through injury.
Speaking ahead of Saturday's home game against Crystal Palace van Gaal said: "We don't need to operate on Marcos, which is a relief. But with this injury it is very difficult that it shall not happen again. And that is the problem.
"We have always a risk that it happens again. Conservatively, it can be less than six weeks. If we have to operate, 12 weeks."
Michael Carrick, recalled this week to the England squad, could again figure as a makeshift centre back but van Gaal has not ruled out entering the transfer market in January for more permanent cover.
"It is clear that a lot of things happen and you can judge better in good times than you can in bad times," he said. "Maybe you are right (about buying in January) but we have to see about that."
And van Gaal admits he feels 'lousy' at the poor start he has made to his time in charge at Old Trafford but is adamant there is no 'quick fix' hinting it could be three years before he turns the club's fortunes around.
"It's not good enough," he said. "I feel, myself, very lousy for the fans firstly, but also for the board because they have a great belief in me and my players and my staff and when you have 13 points out of 10 matches you are not doing well.
"But we are in a process and that process shall take more than one year. It shall take three years, I hope, because that is always dependable from a lot of aspects, but I think it shall be and we shall make it happen that we shall win a lot of matches in a row."
And the Dutchman, who signed a three-year deal in the summer, claims it's not only in defence where he has injury problems ahead of the visit of Neil Warnock's side.
On-loan striker Radamel Falcao has missed the last two games after receiving a kick in training and he too looks set to sit out the match.
"I feel very lousy for the fans firstly, but also for the board because they have a great belief in me and my players and my staff."
Louis van Gaal
Falcao has scored just one goal since his deadline-day move from Monaco but van Gaal is refusing to get too downbeat on the Colombian's return in front of goal until he has regained full fitness.
"I cannot judge him now because now he is injured," van Gaal added.
