Reports in the Spanish media suggested that the Colombian striker was keen to leave the club amid concerns around the tax laws in the principality.
After paying €45million for Falcao in May, Monaco gave him a €14m annual salary, but new laws will see his take-home sum cut in half.
Spanish newspaper Marca claimed that the former Atletico Madrid striker had been offered to Real Madrid, however, the 27-year-old rejected the story.
"I am very happy at Monaco and I believe in the project," he said. "I feel good at the club and I'm staying here."
Falcao has started his career in France well, finding the net on his debut against Bordeaux and opening the scoring in the 4-1 victory over Montpellier on Sunday.
The club were also quick to deny the reports, posting on their official Twitter account: "The club would like to categorically deny rumours about a sale of Falcao.
"As a reminder, the Colombian attacker has signed with ASMFC for five years."
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.