Falcao made the move to Old Trafford on transfer deadline day and rumours were rife that the Colombia striker had been drafted in on loan from Monaco to fill in for Van Persie while the Dutchman undergoes a procedure on a knee injury.
But Van Persie insisted on Tuesday that he does not need an operation and claimed that he is looking forward to playing alongside Falcao.
"I am impressed that people come pick up the imagination to invent something like that," Van Persie told Fox Sports NL when asked about suggestions that he would have to undergo surgery.
"I do not know where it comes from and can say with my hand on my heart that I won't be in a hospital to have the operation."
Van Persie, who signed for United from Arsenal in 2012, has started the last two Premier League games for Louis van Gaal's side but has been substituted in both without scoring.
Falcao's arrival could limit the 31-year-old's starting opportunities but he said he is motivated by the competition.
"I welcome his arrival, he makes us better. At a top club you should always go for the best, which also fits my philosophy," he added.
"Falcao must fight for his place, as I also have to. We must duke with Wayne Rooney and James Wilson to see who's playing."
Van Persie is currently on international duty, with Netherlands facing Italy in a friendly on Thursday before beginning their Euro 2016 qualifying campaign in the Czech Republic.
