Argentina international Messi joined PSG from boyhood club Barcelona in August and the 34-year-old has scored six goals in 16 games in all competitions for the Parisian side.
Defender Juan Bernat, back-up goalkeeper Sergio Rico and midfielder Nathan Bitumazala were the other players who returned positive tests.
Mauricio Pochettino's team are top of the Ligue 1 standings on 46 points after 19 matches, 13 points above second-placed Nice. They are due to travel to third-tier club Vannes tomorrow for a French Cup round-of-32 match.
Pochettino said he was unsure if Messi, who is currently in Argentina, would feature in their next league game, at Olympique Lyonnais on Jan. 9.
France reported 219,126 new confirmed COVID-19 cases in a 24-hour period over the weekend, the fourth day in a row that the country has recorded more than 200,000 cases.
France has joined the United States, India, Brazil, Britain and Russia in having had more than 10 million cases.
PSG also said that Brazil forward Neymar, who is out with an ankle injury, was about three weeks away from returning to training.
Photo: PHOTOSPORT Lionel Messi