Falcao, who must pass a medical, has been linked during the summer with a move away from the French Ligue 1 side, whom he joined for £50m in May 2013.
Falcao has scored 11 goals in 20 appearances for Monaco, and has two in three games this season.
The deal includes an option for United to buy the striker for 55m euros (£43.5m) at the end of the loan spell.
Louis van Gaal’s side, like most of Europe’s top clubs, had been made aware of Falcao’s availability over the past 48 hours.
Premier League champions Manchester City opted against pursuing a deal as they could not make space in their squad for the Colombian.
Falcao had also been linked with Arsenal, as well as Italian champions Juventus and Spanish side Real Madrid.
Rumours of a move to European champions Real gathered pace on Saturday when a comment on his official Twitter account, which was later deleted, read: “Dream come true #halamadrid”.
In a later post Falcao indicated the first tweet was fake, saying: “I have to refute a story about a tweet I have not done and that is a photo montage.”