The Monaco attacker had surgery on Saturday to address a torn anterior cruciate ligament in his left knee, an injury he picked up on Wednesday in the Ligue 1 club's 3-0 Coupe de France victory over Monts d'Or Azergues.
"I am convinced that everything is going to be OK and I will be able to give the best I have (at the World Cup)," Falcao said, during a press conference held at a Porto hospital.
"When they told me at first that I had a serious injury and there was only a small chance of participating (in Brazil), I didn't care whether it was 10, 20 or 50 per cent.
"I was counting on that chance.
"This is my hope and with this dream I will work every day. A lot has to do with my emotional state. This is something that I can manage each day."
Falcao said he understands he is short of time, with Colombia's first group match at the pinnacle event against Greece in Belo Horizonte on June 14.
"Without a doubt, I think that time is not on our side but I think the possibility exists and that is where I find hope," the 27-year-old said.
"I try not to think about whether (recovery) will be two, three, four months, but rather to focus on the day to day."
But Falcao also said he would not be rushing his recovery in order to board a plane to Brazil, but with a likely six-month period required to overcome such a serious injury, that puts his squad spot in significant doubt.
"I can't skip any steps," the former Atletico Madrid and Porto striker said.
"The main thing is that I can respect the recovery steps for a good rehab.
"At no time will I pressure myself or the capacity of my knee to recover."
Falcao is expected to check out of the Trinidade de Oporto hospital on Tuesday and remain in Porto for three weeks to continue his recovery.
Thank you for reading 5 articles this month* Join now for unlimited access
Enjoy your first month for just £1 / $1 / €1
*Read 5 free articles per month without a subscription
Join now for unlimited access
Try first month for just £1 / $1 / €1