The 27-year-old sustained anterior cruciate ligament damage in his left knee during Monaco's 3-0 victory over Monts D'or Azergues in the Coupe de France on Wednesday and could be out for the rest of the season.
"I'm not going to deny my sadness is huge, my heart is broken," he wrote on his Twitter page.
"But I'm clinging to the smallest hope, and that's enough to think that my World Cup dream is still alive. God makes possible the improbable, I believe in Him!!!
"I want to thank all of you for so many messages of support in the last few hours, I'm really touched by the warmth that's been shown to me from all over the world."
Falcao joined Monaco from Atletico Madrid last summer.