UKA chairman Ed Warner said there was "nothing to hide" over the successful London bid but was warned by a "very senior IAAF person" about envelopes of cash being handed out before the vote.
The Daily Mail claims two witnesses recall Coe warning the bid team.
But the IAAF said Coe had "no knowledge of bribes being offered or received".
It added in a statement: "As Coe and Ed Warner discussed on BBC Radio 5 live Sportsweek, there was rumour piled upon rumour in the days leading up to the bid as is often the case on these occasions."
Coe has repeatedly denied he knew about major corruption within athletics during his time as IAAF vice-president, a post he took in 2007 before being appointed president in August last year.
The second part of an independent World Anti Doping-Agency report into the IAAF said "corruption was embedded" within athletics' world governing body.
In addition to allegations of covering up systematic doping in Russia, the report's authors also suggested the awarding of other World Championships should be investigated.
Coe led London's final presentation bid before the vote in 2011, when London beat Doha 16 votes to 10. The Qatari capital was awarded the 2019 edition.
Warner told the culture, media and sport select committee this week that Doha's two bids have been referred to the IAAF's ethics commission.
But the Qatar Athletics Federation has always denied any wrongdoing during the voting process
The second part of an independent World Anti Doping-Agency report into the IAAF said "corruption was embedded" within athletics' world governing body.
In addition to allegations of covering up systematic doping in Russia, the report's authors also suggested the awarding of other World Championships should be investigated.
Coe led London's final presentation bid before the vote in 2011, when London beat Doha 16 votes to 10. The Qatari capital was awarded the 2019 edition.
Warner told the culture, media and sport select committee this week that Doha's two bids have been referred to the IAAF's ethics commission.
But the Qatar Athletics Federation has always denied any wrongdoing during the voting process
