The Colombia international moved to Old Trafford on a season-long loan from Monaco on transfer deadline day, with an option to make a permanent move at the end of the campaign.
The deal represented something of a risk for United, with Falcao having only recently recovered from a serious knee injury which caused him to miss the second half of last season as well as the World Cup in Brazil.
Falcao netted his first goal for the club in Sunday's 2-1 Premier League win over Everton, and the 28-year-old is grateful to United for bringing him to England.
"Manchester United were the ones that gambled on me when it was very difficult to do so," he told Onda Cero.
"It has been a very difficult year, not just in my professional career.
"What happened at the start of the year affected me in every way, but it has also helped me grow as a human being, to think about many things, to mature.
"In terms of my thoughts and plans for the future, I want to stay at Manchester United for the coming years.
"I am still going to need some time to be OK, but I am feeling very comfortable and very good on the pitch, in harmony with myself, with the ball and with football."
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