A Danish newspaper claims it has been confirmed to them by Europol sources - but stressed Liverpool are under no suspicion.
Liverpool won the match at Anfield when Debrecen goalkeeper Vukasin Poleksic parried a shot from Fernando Torres, which fell to Dirk Kuyt.
However, Poleksic was allegedly paid to ensure there were more than two goals in the match, which he failed to do as Liverpool won 1-0 despite having seven shots on target.
A second Champions League tie involving the Hungarian side in the group stages, a 4-3 defeat by Fiorentina, has already been investigated by German police.
The 30-year-old has since been banned for two years by UEFA for failing to report an approach from match-fixers and failed to overturn the suspension at the Court of Arbitration for Sport last year
The FA said in a statement on Monday: "The FA are not aware of any credible reports into suspicious Champions League fixtures in England nor has any information been shared with us."