Arsene Wenger says the Sporting Chance clinic have a duty to name the player who allegedly says he accepted a £50,000 bribe to 'fix' a match.
The Football Association are to investigate a report by The Independent that a player with a serious gambling problem agreed a deal with a bookmaker to write off his debts.
The match is said to have been played in the last two years "in Britain", but the report did not name the player nor state which league or home nation was involved.
The player, it is claimed, agreed to get himself sent off and persuade three team-mates to pick up cautions in a specific match.
The case was disclosed in a seminar on gambling at the Sporting Chance clinic in Hampshire this week, and the clinic's chief executive Peter Kay confirmed he was aware of the issue.
"He has to come out really with it because I don't think it's fair," said the Gunners boss.
"If there is a confidentiality problem, then he (Sporting Chance Clinic) shuts his mouth completely.
"I believe it throws suspicion into our sport that is unfair unless you name the guy."
Wenger maintained: "I don't think it is a problem. However, match-fixing is horrible. I never could believe that it could happen here.
"You want justice to be done. If you want to win the game and you know that the matches are fixed, the sport has no chance.
The Arsenal boss, though, believes such cases are not rife within the English game.
He said: "It [gambling] is part of the culture, but it did not stop this country to have a rather clean league.
"Until now I never had a feeling it was not fair - but if that is proven that the guy has sold the game for £50,000 it is a big surprise to me.
"I think it is a one-off."