Redgrave is a 1-3 favourite to be afforded the honour, but legendary athlete Bannister, the first man to run a four-minute mile, is closing in as second favourite at 4-1 after originally being considered a rank outsider.
When we polled Eurosport-Yahoo! readers last year, Bannister - who broke the famous mile mark in May 1954 - was one of the names repeatedly suggested, and his falling odds reflect that people would love to see the 83-year-old neurologist honoured.
"Sir Roger started as an underdog and he's caught us by surprise, slowly creeping up the market to second favourite," Ladbrokes spokesperson Jessica Bridge said, but cautioning that Redgrave is still the likeliest choice.
"Sir Steve is our greatest Olympian and all the money's suggesting he'll do the honours."
However, organisers have strongly suggested that the final choice will be something out of the ordinary, with a spokesman saying that the decision "will surprise the billion people watching around the world".
Such a cryptic statement suggests that there may be something in suggestions that David Beckham may be the man. The former England captain, once out at 33-1, has come down to 10-1 third favourite after a flurry of betting, with Dame Kelly Holmes and Tom Daley not far behind at 14-1.
"Since Becks brought the flame back from Greece we've had an unprecedented amount of bets on him, which has seen his odds slashed from 33/1 to 10/1. It will be a huge upset if Redgrave doesn't get the gig though," Bridge added.
Daley Thompson, who is at 16-1 shot alongside Chris Hoy, admitted that he doesn't even know how the decision is made.
"I couldn't even tell you who are the people who choose who does it. It would be interesting to know what the criteria is," he told the Daily Mirror, adding in his own inimitable style, "but if it's a very handsome working-class bloke than I suspect I may well get the nod. I'd love to do it - but who wouldn't?"