It is understood that the 28-year-old is due to arrive in Manchester ahead of a move.
A deal for one of the most sought-after strikers in the game would see United beat both Manchester City and Arsenal to the South American's signature.
Meanwhile, United's Mexican striker Javier Hernandez flew to Spain earlier on Monday ahead of a proposed loan move to Real Madrid with an option for the Spanish club to buy the 26-year-old at the end of the season.
United had not been expected to make another major signing after reaching an agreement to bring in Daley Blind from Ajax, but the chance to land Falcao has proved too tempting especially with Hernandez and possibly Danny Welbeck leaving Old Trafford.
It would mean another season without Champions League football for Falcao, but that is not something that has concerned him in the past - indeed not since he was with Porto in the 2009-10 season has he played in UEFA's top club competition.
The player had been looked at by Arsenal and City but both clubs had baulked at the financial investment needed in terms of the loan fee and his wages.
Meanwhile, United are also expected to confirm the completion of Blind's move after agreeing a deal for the 24-year-old, who can play in defence or midfield.
Blind, who would sign for £13.85million, was a key part of United manager Louis van Gaal's Holland team in the World Cup.
The player told Fox Sports at the weekend: "I am enormously proud to be at one of the biggest clubs in the world. Everything has happened so suddenly. What exactly has gone off in the background, only my agent and my dad know, but it was a very exciting and tense time."