Mayor Ken Livingstone believes a single vote could ultimately decide whether London or Paris gets the 2012 Olympics.
Paris is seen as the favourite to beat London, Madrid, New York and Moscow in the election in Singapore on 6 July.
"I don't think it will be clear until they announce the result. I think it will be a handful of votes between us and Paris on the last ballot.
"It is not impossible that this vote will be decided by 55 votes to 54," Livingstone told BBC Sport.
"I suspect it could very well be decided on the last day, just by the quality of the presentations of the rival cities."
The all-important votes could be won or lost in the final few seconds of lobbying, says Olympic heptathlete Denise Lewis, who like Livingstone, is flying to Singapore to help with last-minute campaigning.
"It's like going into a head-to-head competition," said Lewis.
"The difference between winning and losing is very small and it could be a matter of the last 30 seconds. We're so close, it really could go either way."
Livingstone leaves on Friday to join bid leader Lord Coe and number two Keith Mills, who went out early to fine-tune London's presentation.
All five bid teams will spend the days before the vote trying to win over International Olympic Committee members.
The cities will all sign contracts to host the Games before the vote, and those of the four losers will be shredded after the winner has been announced.
Livingstone believes Coe and Prime Minister Tony Blair, who will spend 48 hours in Singapore before the vote, could be the key to making sure London is the last city standing.
"Seb Coe has been exactly the right person in this last year," he said.
"We are the only one of the five cities where the bid is being led by a gold medal-winning athlete.
"I often have to be briefed about who it is I'm meeting, but he knows the lot.
"All London's plans have been drawn up by sportspeople, so every single aspect has the sporting mentality stamped all over it. I think that's one of our biggest selling points."
Livingstone added: "I think another of our strongest plus points in these last few days is the Prime Minister's ability to work a room. People really relate to him.
"He had an enormous impact among IOC delegates when he was at the Athens Olympics last summer.
"It will be really interesting to see how Tony Blair and (French president) Jacques Chirac work in those last few days."
Livingstone admits London's transformation from an outsider to a potential winner since the bid was launched in 2003 has surprised him.
"Four years ago, I thought with all the fiasco of the Dome, Wembley and Pickett's Lock that we didn't have a chance of winning this, but we'd get resources for London out of it," he said.
"And then in this last year it's been quite clear that the bid we've put together is so attractive for the whole international sporting community that we're in there with a chance of winning.
"Basically we're fighting for every vote we can now get.
"I think it would be absolutely electrifying in 2012 for the whole nation to have that focus on a brilliant Games, given the enthusiasm of the British people for sport.
"When I went to Athens last summer, there were more Brits than any other nationality except the Greeks, and that was noticed by the IOC members as well."