Sir Alex Ferguson revealed the news at his pre-match press conference for Saturday's lunchtime clash with Everton.
While Anderson's injury is described as not serious, Carrick faces yet more frustration, having made only one league appearance so far this season.
Coming on as a 74th minute substitute for Paul Scholes in the 3-0 win against West Ham last month, the England international has had to watch from the sidelines as United beat Newcastle and drew with Fulham in the opening two games of the season.
But now he faces even more time out of the action, with Ferguson saying: "He should be out for three weeks, maybe a month at most. Carrick was an unused substitute during England's recent Euro 2012 double-header but has not started a game for United since the Community Shield triumph over Chelsea at Wembley last month. "He has had a problem with his Achilles," added Ferguson. "It has been bothering him for a few weeks now, so we sent him to a specialist in London yesterday and he had an injection. "We are hoping that settles it down. "We have been considering what we should do about it. The best thing is to get it treated and out of the road now because everyone in the squad is fit." Anderson was withdrawn as United's reserves beat Stockport reserves 2-0 in the Manchester Senior Cup. The Brazilian hasn't played a first team game since February after suffering cruciate ligament damage.