"The NCA is working closely with the Gambling Commission and the Football Association," the NCA said in a statement.
"The focus of the operation is a suspected international illegal betting syndicate."
The Daily Telegraph, which has been investigating match-fixing across the globe for months, reported that the men arrested included three footballers and a former Premier League player who is now an agent.
According to the newspaper, the games are all in the lower leagues. The Telegraph did not specify the relevant divisions but one team is alleged to have been involved in three fixed games.
"The FA has been made aware of a number of arrests in relation to a National Crime Agency investigation," the FA said. "We have worked closely with the authorities in relation to these allegations."
The last major scandal of this sort in England occurred in the mid-1960s when 10 players were found guilty and jailed for conspiring to fix matches.
Among those sentenced were England pair Tony Kay and Peter Swan, and David 'Bronco' Layne, who all played for Sheffield Wednesday.