City of London Police said detectives from its Economic Crime Directorate had assessed information gathered during an investigation by the Daily Telegraph newspaper in September.
"This review of the material has concluded and the decision has been taken to begin a criminal investigation into a single suspected offence of bribery," the police said in a statement. No further details were given.
The Telegraph investigation led to the English Football Association (FA) sacking Sam Allardyce as the national team manager after just one match in charge, ruling he had behaved "inappropriately" when discussing an offer of a lucrative sideline role put to him by undercover reporters.
Police said Allardyce was not part of their investigation, news which he welcomed in a statement.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
"I was always confident that this would be the case as there was no evidence against me," Allardyce said. "I now ask that the Football Association deals with this matter as quickly as possible."
The Telegraph also reported that eight current and former Premier League managers were accused of receiving unofficial payments, known as 'bungs', for player transfers.
The paper said it had agreed to hand over its findings to the FA and the police after undercover reporters discovered "widespread evidence of corruption in the English game" by filming soccer agents boasting about how many managers they had paid off.
(Reporting by Michael Holden; Editing by Mark Trevelyan)
