The Colombian made the move to Old Trafford from Monaco on transfer deadline day last Monday, but is already looking to the future.
In today's press conference, the striker said: "I hope to stay many years at Manchester United and make history at this club.
"When I was at Porto and Atletico [Madrid] I always wanted to improve and I dreamed of playing in a team like this. Now I want to stay here for many years."
Reports have claimed that the Red Devils paid a £6m loan fee to Monaco, who are believed to want around £40m next summer, should United opt to buy Falcao.