The Red Devils moved quickly to secure a deal for the Colombian striker in the final days of the window. The Premier League giants beat the transfer deadline to sign him on a loan-to-buy deal in the early hours of Tuesday morning UK time.
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Having suffered a cruciate knee ligament injury last season, Falcao missed the World Cup with his country but scored two goals in three Ligue 1 games for Monaco upon his return in 2014-15.
And Van Gaal has expressed his happiness at acquiring the 28-year-old, who becomes the club's sixth major signing of the summer following the arrivals of Angel Di Maria, Ander Herrera, Daley Blind, Luke Shaw and Marcos Rojo.
"I am delighted Radamel has joined us on loan this season," the Dutchman told the club's official website. "He is one of the most prolific goalscorers in the game.
"His appearance-to-goal ratio speaks for itself and, when a player of this caliber becomes available, it is an opportunity not to be missed."
Falcao could make his United debut against QPR after the international break, as United goes in search of its first win of the Premier League season.