Monaco did not elaborate on the length of Falcao's absence, while media reports suggested he could be out for six months. The tournament in Brazil kicks off on June 12, less than five months from now.
Colombian Football Federation sources said the 27-year-old, one the world's top strikers, would undergo surgery this weekend in Portugal by a team led by Jose Carlos Noronha, who was cautious about predicting Falcao's World Cup chances.
"Radamel Falcao suffered a lesion of the cruciate ligament in his left knee. Surgery is necessary... the doctors will do everything they can to help Radamel Falcao recover as quickly as possible," Monaco said in a brief statement.
"We only have five months, which is very little, but we'll closely follow his evolution," Noronha told Colombian radio station Caracol.
"Falcao is a very responsible athlete, we're going to operate on him very shortly and we hope he'll be good but we can't say if he'll play the World Cup."
The striker nicknamed 'Tiger', who was looking forward to his first World Cup finals, picked up the injury after scoring his team's first goal in a 3-0 win at amateurs Chasselay Monts d'Or Azergues in a last-32 French Cup game on Wednesday.
"I'm not going to hide that my sadness is huge, my heart is destroyed, but I'm hanging on to a hope the size of a grain of mustard and that's enough to believe that my World Cup dream is still alive," Falcao said.
"God makes the improbable possible, I believe in Him," he added in a message on Facebook. He also Tweeted thanks to fans and friends for their messages of support.
GOOD PARTNERSHIP
Falcao was a key member of the Colombia side that coach Jose Pekerman steered to second place in the South American qualifying group, securing a first finals berth since 1998.
Colombia face Greece, Ivory Coast and Japan in Group C.
Falcao top-scored for Colombia with nine goals in 13 qualifiers and forged a good partnership up front with Teofilo Gutierrez which yielded a tally of 15.
A key element in Colombia's success was Pekerman's faith in James Rodriguez as the most advanced midfielder, feeding the strikers or running into space to score three goals himself.
The combination of Falcao and Rodriguez was not lost on Monaco, who swooped for the pair from Atletico Madrid and Porto respectively when they were promoted and rebuilt for an assault on Ligue 1 this season.
Falcao's absence is also likely to seriously dent Monaco's Ligue 1 title aspirations.
The Colombian, who joined from Atletico during the close season for a fee exceeding 50 million euros ($68.40 million), has scored nine goals from 17 league appearances.
Monaco are second in Ligue 1 on 45 points from 21 games, five adrift of French champions Paris St Germain.
Falcao missed a large part of the 2005-06 season when he played for River Plate in Argentina after rupturing ligaments in his right knee.
($1 = 0.7310 euros)
(Additional reporting by Luis Jaime Acosta in Bogota and Rex Gowar in London; Editing by John O'Brien)
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