The centre-back missed United's opening two games of the season and the latest scan results will come as a major blow to club and country.
United have got off to a faltering start in the English Premier League and Fabio Capello's England team play a friendly with Slovenia before the 2010 qualifier at home to Croatia on September 9
United, stunned 1-0 at Burnley in midweek, will at least welcome back Gary Neville and Nemanja Vidic, who has been out with a calf problem, for Saturday's visit to Wigan.
"It has been established Rio will be out three to four weeks," United manager Alex Ferguson told reporters "Experience is important and the kind of experience that Gary Neville would bring is obviously great experience.
"Vidic's performance levels since he has been with us have been outstanding. You always want that in your team and we've got that back tomorrow. He has been training with us for 10 days and will be okay."
Despite their shaky start, a long injury-list and only one goal in their opening two games, Ferguson insisted he will not be attempting to sign any new players before the transfer window closes in 10 days.
"I have made the point, this is a good group of players," said the Scot. "We have dominated the last two games of football although we would have liked to have scored more goals.
"Our ratio of goals last season dropped for the first time, it was our poorest for 15 years in terms of goal difference.
"That was the issue last year and it's going to be the issue this year unless we step up to the mark. I think we can do that. We can win games alright, this is a good squad of players."
(Editing by Alastair Himmer)