The Monaco striker was caught in a sandwich between goalkeeper Yoann Cardinale and defender Paul Baysse just before half-time in the 4-0 defeat at Nice.
"Radamel was caught in a sandwich and stayed on the ground with head pain for some time," said Monaco coach Leonardo Jardim.
"He's gone to hospital for tests."
He didn't appear for the second half and went to the Princesse Grace hospital in Monaco for scans. Due to head trauma protocols, Falcao must remain in hospital overnight.
Monaco next play Angers on Saturday before hosting Bayer Leverkusen in the Champions League on Tuesday.
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