The striker joined the Reds on transfer deadline day in September, initially on a temporary basis with an option to make the move permanent.
However the Colombian failed to discover the form which made him one of the most feared forwards in world football and managed just four goals in his 29 appearances for United.
Louis van Gaal paid tribute to the 29-year-old’s professionalism during his time at the club.
Van Gaal told United’s official website: “Falcao is a top professional and a good human being. On behalf of myself and everyone at the club I would like to wish him well for the future.”
Falcao is unlikely to be the only player to depart Old Trafford this summer as Van Gaal attempts to construct a squad capable of challenging for the Premier League title next season.
David de Gea has been heavily linked with a move to Real Madrid while the futures of Robin van Persie and Angel Di Maria are uncertain after both endured disappointing seasons with the club.
Van Gaal, who will fly to his villa in Portugal this week after a draining year, confirmed all decisions have been made regarding transfer targets and he expects to be in constant contact with United executive vice-chairman Ed Woodward while he is away.
“I go on holiday but I have contact with Ed Woodward every day as he is doing the negotiations,” Van Gaal said. “It shall be a heavy (busy) holiday.”
“The transfer period is a process. I know which players have to go and which have to come but you never know if they go or if they come because it will depend on a lot of aspects.”