United boss Sir Alex Ferguson confirmed the news at his press conference on Friday morning.
"Kagawa is out for three to four weeks with a knee injury. Playing on Tuesday (against Braga in the Champions League) didn't help. It is disappointing," the Scot said.
On a positive note, Ferguson reported that defender Chris Smalling was back in training following a broken metatarsal.
"That is a big bonus for us," Ferguson added.
"Phil Jones should start (training) a week on Monday as well."
Jones suffered a knee injury in training last month.
Meanwhile, Chelsea have reported that Frank Lampard will miss at least two weeks with the calf injury he picked up in the midweek Champions League defeat to Shakhtar Donetsk.
The 34-year-old midfielder had to be substituted midway through the first half as the European champions lost 2-1 to the Ukrainian side.
Chelsea boss Roberto Di Matteo on Friday confirmed that Lampard would be out for a couple of weeks, meaning he will initially miss the back-to-back games against United including Sunday's Barclays Premier League clash at Stamford Bridge.
Ferguson welcomed the decision of United defender Rio Ferdinand to put out a statement, along with his brother Anton, on Wednesday night in which they clarified their position over the issue of racism in the game.
Both players refused to wear T-shirts in support of the Kick It Out anti-racism campaign group prior to their respective clubs' matches last weekend, with Rio's decision to boycott the campaign initially drawing an angry reaction from Ferguson.
Their joint statement hailed the "fantastic work" done by Kick It Out, but still expressed their disappointment at the actions of the Football Association in relation to its handling of the John Terry case. The Chelsea defender was given a four-match ban for racially abusing Anton Ferdinand.
"Rio and Anton have done the right thing with that statement," Ferguson said.
"It has been bubbling along and the racism issue has not completely gone away. I am sure the PFA are doing their best. The FA, UEFA and FIFA need to do more."