Radamel Falcao's father claims Arsenal made a late bid to sign his son and that Liverpool, Manchester City and Juventus also wanted the Colombian.
Falcao, 28, was linked with European champions Real Madrid before sealing a season-long loan at Manchester United.
"The clubs that showed an interest were Juventus, Man City, Liverpool and Arsenal at the last minute," Radamel Garcia told a Colombian radio station.
"Real Madrid chose not to. Thank God everything came through with United."
Having joined Monaco for £50m from Atletico Madrid in June 2013, Falcao scored 11 goals in 20 appearances for French Ligue 1 side last season, before sustaining a knee injury in January that kept him out of the World Cup.
"He was very happy at Monaco but unfortunately, they changed the original plan and decided to sell the players they had bought," Garcia said.
Falcao, who will wear the number nine jersey at United, is expected to make his debut at Old Trafford against QPR on 14 September.
Also in the transfer window United signed winger Angel Di Maria from Real Madrid for a British record transfer fee of £59.7m, midfielder Ander Herrera from Athletic Bilbao for £29m, Southampton defender Luke Shaw for £27m, Argentina international defender Marcos Rojo for £16m from Sporting Lisbon and Netherlands international Daley Blind from Ajax for £13.8m.