The Monaco striker made a surprise temporary switch to Old Trafford on transfer deadline day but is keen to be part of Louis van Gaal's plans in the long term.
"I am very happy to be here," he told the press. "It's a big challenge for me, for my career, I always dreamed of being at a club like Manchester United.
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An injury had kept Falcao out of Colombia's World Cup campaign this summer but the striker is adamant he is fit and ready to star for United and, hopefully, earn a permanent move.
"I feel well," he added. "I started to play two months ago with Monaco and I improved a lot in the last month. I score goals; that is important to strikers. I am confident with my physical form and I am comfortable with my knee.
"It's a new process, a new team. Some new players have come this summer and I think Manchester United are building a big team for this season and future seasons and I would like to be here and be part of this project.
"I'm fine, I'm content. The group welcomed me well [and] playing with talented players makes things easy to settle in.
"I don't think I will need too much time because Manchester United has very good players and, for that reason, it's easier to play football and I'm used to life in cities like this. I don't think I'll have a problem adapting.
"I see the club in a period of getting to know each other. There are a lot of new faces but there are lots of top players who are very intelligent so they are bright enough to fit into the manager's plans. However he sets us up, we will be able to put that into practice on the field."
After finishing seventh in a disastrous 2013-14 season, United could not offer the prospect of Champions League football to new signings this summer but Falcao was undeterred.
"It's true, every player wants to play Champions League," he added. "This year it's not possible in Manchester but we are confident of achieving the qualification for next season."
Falcao was understood to be close to joining Real Madrid earlier in the transfer window and even had to delete a Twitter post which included the hashtag #HalaMadrid, but is adamant that he made the right move in the end.
"Everybody has heard about that team," the Colombian said, referring to Madrid. "I don't want to talk about them.
"I am very happy here, I believe in this project, I think United is the best club for my style of play and I am sure of my decision. I am sure that, in the future, it will be great for us."
Daley Blind was the Red Devils' other deadline-day arrival and the former Ajax man says that he is happy to play in whatever position the manager chooses.
"It's important I can play as a defender and a midfielder and I want to play so, if it's in either position, it's up to the coach and I try to do my best wherever I play," the Dutchman added.
Asked about ex-Netherlands boss Van Gaal's approach, the 24-year-old replied: "He's very strict, he's very direct with what he wants, the style he wants for the team. He asks the players to do exactly what the best is to win the game.
"Every game it could be a different style. During the World Cup I mentioned that he reads the games so well and his predictions about the games come true almost every game."