The move comes after speculation linking the Atletico Madrid striker with a move to Old Trafford over the last couple of days.
A report in Spanish newspaper Marca on Wednesday claimed United were in advanced talks over a deal and now there are rumours that the Colombia international could make arrive in a deal that will see Javier Hernandez used as part-exchange.
The 27-year-old has enjoyed another stellar season with 22 goals so far in the Primera Division and Atletico manager Diego Simeone expects interest in his star man.
"Falcao is a great player and it's logical that the best teams in the world are interested in him. It doesn't surprise me," he told reporters on Wednesday.
Chelsea and Real Madrid are the joint second-favourites at 5/1 to sign the striker.