Premier League matches are not reported to be under investigation, and the Football League, which runs the three professional divisions below the Premier League, said it had not been contacted by police.
The English Football Association is working with Britain's National Crime Agency on the investigation.
"Six men have been arrested across the country as part of an NCA investigation into alleged football match fixing," the NCA said in a statement. "The focus of the operation is a suspected international illegal betting syndicate."
At meetings in Manchester this month, the Telegraph said one of the alleged fixers a Singaporean man correctly predicted how many goals would be scored during a match the next day, and offered to manipulate two British matches this month.
The man told the paper's investigator in a video that he would say to a footballer: "You tell me how many goals you can give."
"Either 3-2, 4-1 or zero," he added in broken English. "I say I don't need five. For me four is enough ... if more than that up to you. But my deal is four ... I don't want less than four."
The alleged fixer is heard claiming he has a betting website, stressing: "We can bet (on) those goals."
He also claims he can pay a player about 5,000 pounds ($8,140) to ensure he is booked in the first ten minutes of a match, an indication that the game is being fixed.
The Telegraph said it was approached by an "undercover investigator with links to FIFA, who had been gathering evidence against suspected Asian match fixers offering to operate in Britain."
The alleged fixer said he was connected with Wilson Raj Perumal, the Singaporean who was sentenced to two years in prison in Finland in 2011 for bribing players in the Finnish football league.
"He's my boss," the alleged fixer in the Telegraph investigation said. "Everybody in the world know him."
Now co-operating with authorities in Hungary, Perumal has given evidence which is key to FIFA and law enforcement agencies piecing together the scope of match-fixing plots worldwide carried out by crime syndicates with Singapore connections.
Match-fixing is a growing blight in football, with investigations across the globe raising concerns about the integrity of the sport.
"Everyone really knew that match fixing is endemic in football," Chris Eaton, the former head of security for FIFA, told The Associated Press on Thursday. "And in this (alleged) case there is nothing new in terms of the corrupting method, its internationality or in the core betting fraud purpose.
"What is new is that it shocks England, the home of the game. That shock should be used to galvanize international efforts to regulate and supervised sport betting globally, which is the real motivation for modern match-fixing."
The scale of the corruption in football was highlighted earlier this year when Europol, the European Union police liaison agency, said it reviewed 680 suspicious recent cases of match-fixing.
One of the biggest recent alleged fixing plots was unearthed in Australia where four English players were charged in September in a criminal investigation. Before heading to Australia to play for the Melbourne club Southern Stars, the men played in England's lower leagues.
"It was only a matter of time before the English game was caught up in this global wave of match fixing in football," said Eaton, who is now director of sport integrity at the International Centre for Sport Security.
"International sport, especially football, is in serious trouble with corruption of its competitions."
The last major match-rigging convictions in English football were in the 1960s. Peter Swan, David 'Bronco' Layne and Tony Kay, who were all in or on the fringe of the England team, were jailed for four months and banned for life for corruption.
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Daily Telegraph investigation: http://bit.ly/1dBMFaA
