The Colombia international had scored the opening goal in Monaco's 3-0 victory, but was forced off before half-time after tangling with a home defender.
Falcao looked in considerable pain and had to be carried off the field on a stretcher.
If serious, the injury could keep him out for the rest of the season as Monaco look to make up a five-point gap to runaway Ligue 1 leaders Paris Saint-Germain.
The setback also throws his participation in this year's FIFA World Cup into doubt, where he is set to be one of Colombia's key players.
A Monaco statement said: "The player Radamel Falcao was hit in the left knee.
"We must await further tests in the coming days to clarify the exact nature of the injury."
Colombia were drawn in Group C for the World Cup, along with Greece, Ivory Coast and Japan.
