The pair were playing on opposite sides in Colombia's first practice in the north-central city of Santa Fe since arriving in Argentina for the regional tournament which starts Friday.
Television footage on Fox Sports showed striker Rodallega, charging down a pass outside the box, caught Ospina in the face with his shoulder when the goalkeeper came out to kick the ball clear.
"He has a fracture and we're going to establish the severity of the injury. We're not sure if the player can [play the Copa] or not," Colombia team doctor Carlos Ulloa said.
"If it's a small fissure we'll pass him fit, if not, then we won't," he told Fox Sports.
Ospina, who lay motionless after the impact, received pitchside treatment before being taken from the training ground by a member of the coaching staff who held a dressing pad to the keeper's face.
No-one with the team was immediately available to confirm the severity of the injury.
Colombia are in Group A with hosts Argentina, Bolivia and Costa Rica. They meet Costa Rica in their opening match in the north-western city of Jujuy on Saturday.
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