Rooney, who also broke a bone in his foot during Euro 2004 and just before the 2006 World Cup, had the injury confirmed by a hospital visit, the club said on its Web site (www.manutd.com).
''The England striker was substituted at halftime after being caught by Michael Duberry as the Royals defender tried to block a shot,'' United said.
''Although Rooney did briefly play on after receiving treatment, Sir Alex Ferguson opted not to risk any further injury by fielding him in the second half, and a post-match hospital scan has validated that decision.
''The result of Rooney's scan has worsened a frustrating opening day of the season, and as yet there is no timescale of his recovery.'' Rooney's injury, which could well rule him out for around three months, is a massive blow to both club and country, who were already facing a real battle to qualify for Euro 2008.
The striker, United's joint top scorer last season with Cristiano Ronaldo on 23 goals, is also a vital part of England's bid to reach the European Championship finals in Austria and Switzerland.
England, under Steve McClaren, are lying fourth in Group E with 14 points from seven games after a lacklustre campaign in which the coach has faced considerable criticism.
They trail Russia with 15 points, Israel on 17 from eight games and leaders Croatia, who also have 17.
The injury woes of the hugely talented striker were key reasons for England's exits at their last two tournaments.
Rooney broke a metatarsal in the early stages of England's quarter-final loss to hosts Portugal in 2004 and his lack of form after another injury was a big factor in their defeat by the same opponents at the same stage of last year's World Cup.
Ferguson had been hoping for the best immediately after Sunday's season opener at Old Trafford, telling Sky Sports News: ''It was painful, he couldn't really stand on it.
''I don't think it's the same injury as the last, obviously we hope and pray that that's the case.
''But it was sore and it started to swell so there was no point in trying anything for the second half.'' However, the hospital confirmed the worst fears for Ferguson, who had underlined Rooney's absence for the second half.
''I think the loss of Wayne Rooney was obvious,'' the Scot said.
''If he'd have been on the pitch I think we would have won it.''