Van Persie damaged ligaments during United's 2-1 defeat by Swansea City last weekend and left the Liberty Stadium on crutches with his right foot in a protective boot.
"It is an ankle problem. I do not think it is very heavy but ankle problems can take a long time," manager Louis Van Gaal told a news conference on Friday. "It is not one or two weeks."
Dutch striker Van Persie has struggled for fitness and form this season, scoring 10 goals in 24 appearances, and could be replaced by Falcao.
However, the Colombian, who joined United on a season-long loan from AS Monaco for 6 million pounds, has failed to live up to expectations, scoring just four goals in 19 appearances.
United have the option of paying around 43 million pounds to make Falcao's move permanent at the end season, but Van Gaal did little to boost the 29-year-old's confidence when asked if he would start against Sunderland.
"The possibility is higher (that Falcao will play) of course because when a striker is away another has to play," he said. "We will have to wait and see."
United could climb above Arsenal into third in the Premier League table if they beat Sunderland at Old Trafford as the Gunners host Everton on Sunday.
Van Gaal admitted it was "a rat race between five clubs" to finish in the top four and said it would be a "big disappointment" if United failed to qualify for the Champions League.
"Our process is continuing and always we are going better," the Dutchman said. "In spite of the many injuries we have had in the first half of the season and also in spite of the criticism of our defence, we are there.
"We are still in the FA Cup so I am pleased with the results, but not with a lot of the performances of my team.
"When you see our matches we are mostly the better team, but the better team is not always winning."
(Editing by Ed Osmond)
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