The Monaco striker was heavily linked with a move to the Red Devils early on Monday morning after being linked with Arsenal and Manchester City over the past week.
Falcao, who has previously spoken of his desire to join Real Madrid, will reportedly cost United £20m on a season-long loan deal, including an option to make the move permanent.
The 28-year-old has scored 11 goals in 20 Ligue 1 appearances following a big-money move to Monaco from La Liga winners Atletico Madrid last summer.
United have already brought in midfielder Ander Herrera, England international Luke Shaw, Argentine Marcos Rojo and British record £59.7m signing Angel di Maria.
However, Louis van Gaal has been criticised for failing to land a world-class centre-half, prompting Phil Neville to take to Twitter earlier on Monday afternoon.
I'm excited – nice to add a centre back now!!! pic.twitter.com/6JF3ekwOdZ
— Philip Neville (@fizzer18) September 1, 2014
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