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 Poleksic 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."
The European police agency (Europol) revealed on Monday that a wide-ranging investigation into match-fixing has uncovered more than 380 suspicious matches, including World Cup, European Championship qualifiers and two UEFA Champions League games.
Europol's chief Rob Wainwright said the investigation found "match fixing activity on a scale we have not seen before", and revealed the probe uncovered around £7m in betting profits and £1.27m in bribes to players and officials and has already led to several prosecutions.
The investigation identified about 425 corrupt officials, players and serious criminals in 15 countries and 50 arrests have been made so far.
Wainwright also stated that the involvement of organised crime "highlights a big problem for the integrity of football in Europe" with Asian gambling cartels identified as part of the match fixing operation.
European football's governing body, meanwhile, stressed it had a zero-tolerance approach to match-fixing.
It added in a statement: "UEFA is aware of the statements made by Europol regarding the alleged match-fixing that has taken place in various football competitions and expects to receive further information in the coming days.
"As part of the fight against the manipulation of matches, UEFA is already co-operating with the authorities on these serious matters as part of its zero-tolerance policy towards match-fixing in our sport.
"Once the details of these investigations are in UEFA's hands, then they will be reviewed by the appropriate disciplinary bodies in order that the necessary measures are taken."