"I am well aware of the extent of the problem in Russia," Eaton told Reuters in an interview.
"You can just go through FIFPro's so-called 'Black Book' to find out how deep the problem is in eastern Europe and Russia."
"There are three basic elements in the fight against corruption and match-fixing," Eaton said at the International Sport Security Conference (ISSC), held this week in Qatar.
"First, you need good government structure. Then you have to have information or what we call the intelligence operation. And finally, you must implement that in practice.
"The Russians have a good opportunity to get all these elements together," said the Australian, who recently made his first visit to Russia.
"I've had a good dialogue with Russia's chief of referees [Italian Roberto Rosetti] and I plan to make regular visits to (the) country in the future."
"You would have to be very naive to think that match-fixing only exists in certain parts of the world, like western Europe," he said.
"Criminal gangs have infiltrated the game around the world to the extent that is making FIFA very concerned.
"I don't think any country in the world can be totally immune from it. Right now, FIFA's priority is Africa and South-East Asia but Russia will also be closely looked at, especially after being awarded the World Cup in 2018.
"Don't forget, criminals have had a 10-year head start on law-enforcement people in this particular area," added Eaton, who is leaving FIFA in May to take the job of director of sport security for the Qatar-based Centre for Sport Security.
"I think in soccer FIFA has established a pretty good structure to fight corruption and match-fixing," he said.
"In my new job I will try to help other sports, such as basketball and volleyball, to deal with this problem because the very same people who fix soccer matches also try to influence the outcome in other sports as well and my job will be to stop th
