"There is no amnesty, not today," Eaton told the Sunday Mail.
He is in Harare to meet football officials.
"We have got zero-tolerance on match-fixing and we have to understand that this is now a big problem facing the sport."
Eaton dismissed local newspaper reports that anyone found guilty would be pardoned and sent for rehabilitation instead of being banned.
"We want our football to be clean because criminals take advantage of the sport. We didn't invite gambling and now, because of gambling, we have got criminals."
Zimbabwe is under investigation by FIFA over an alleged match-fixing scam in Asia involving the national team.
Former Zimbabwe Football Association (ZIFA) chief executive Henrietta Rushwaya sent the national team to play unsanctioned friendlies in Thailand, Syria and Malaysia two years ago and a betting syndicate allegedly fixed the results.
Rushwaya was fired last October.
She is also said to have cleared former league champions Monomotapa to travel to Malaysia masquerading as the national team.
Last month, ZIFA suspended three board members, including a former national team player and a former referee, for alleged involvement in match-fixing.