Prime minister indicates anything is possible, and says corruption inquiry relating to Qatar bid should take its course

David Cameron has indicated that the World Cup bid for 2022 could be thrown open, saying anything could be possible after the allegations of corruption about the Qatar bid.

The prime minister, who spoke of unhappy memories when he took part in the final stages of England's 2018 bid at the end of 2010, said the inquiry into corruption relating to the Qatar bid should be allowed to take its course.

Speaking during a Q&A in Newark before Thursday's byelection, the prime minister said: "We will see what happens with this inquiry into the World Cup. And who knows what the chances may be for the future."

His remarks may fuel speculation that he believes that the England bid for 2018 could be reopened, because the 2018 and 2022 World Cups were decided at the same meeting, which he attended with the Duke of Cambridge. But Russia is unlikely to lose the 2018 World Cup and England could not host the 2022 tournament because it was designated for a non-European country.

Asked whether the 2022 bid should be reopened in the wake of the allegations in the Sunday Times, Cameron said: "There is an inquiry under way, quite rightly, into what happened in terms of the World Cup bid for 2022. We should let that inquiry take place rather than prejudge it. My memories of that bidding process are not happy memories in terms of the way the whole thing was arranged and the role of Fifa."

He also indicated that he would not support stripping Russia of the World Cup in 2018 after its annexation of Crimea. "As for the future of the World Cup in 2018, generally speaking we should try to keep sport and politics separated. We should use that as a rule. In extremis there are occasions – Zimbabwe, South Africa – where it is right to not take part in sporting occasions."