"If anything comes out of that nature, then clearly all those bids will be called in," Coe told BBC Radio 5 live.
"In a bidding process, every sport has rumour piled upon rumour."
London won the 2017 vote 10-6 in November 2011. Qatari capital Doha was subsequently named as host for the 2019 event in November 2014. Qatar has denied any wrongdoing in either bid.
A World Anti-Doping Agency independent report published on Thursday found that "corruption was embedded" within the International Association of Athletics Federations, the sport's world governing body.
Coe became IAAF president in August after eight years as vice-president.
A French criminal investigation into Coe's predecessor, Lamine Diack, and other leading IAAF officials is looking at whether there was any financial wrongdoing in bids for seven World Athletics Championships from 2009 onwards.
Co-author of the Wada report Richard McLaren said he suspected bribes might have been paid, adding "we don't know that for sure - that's why we want a further investigation and the police are pursuing that".
