The England international was injured in an accidental collision before Sunday's defeat at Liverpool.
"He got a cut down the middle of his forehead which was completely accidental," United boss David Moyes told Sky Sports.
"I've got to say it's quite a bad cut and it will probably keep him out for a few weeks."
While the FA confirmed on Sunday night that Rooney was out of England's World Cup qualifiers, the United manager explained more about the injury to reporters at his post-match press conference.
"It [the cut] is right down the middle of his forehead where you haven't got much skin; it'll take a wee while to heal. I'd be guessing he'll be out for three weeks," said Moyes.
"It was an accident. He turned around and Phil Jones had just lifted his boot and it caught him right down the middle of his forehead."
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