Mr Burnham referenced the notorious Oxford University dining club - which counts David Cameron and Boris Johnson as former members - as he insisted the contest was "still there to be won".
The ballot closes at 12:00 on Thursday, with the winner announced on Saturday.
Mr Burnham is up against Jeremy Corbyn, Yvette Cooper and Liz Kendall.
Left-wing MP Mr Corbyn has gone from outsider to bookies' favourite, triggering a row within the party and warnings from senior figures, including Mr Blair, about him winning.
In his final public speech of the campaign, Mr Burnham will tell party supporters in Watford Labour must not fight "the battles of the past".
The shadow health secretary will pledge to make Labour "a proper opposition", adding: "No longer will Labour look like a pale imitation of the Tories."
There has been criticism of the process by which people can take part in the vote, with claims of infiltration by non-Labour supporters. People are able to pay Â£3 to become a registered supporter, entitling them to a vote in the contest.
There has been a huge increase in the party's electorate, with more than 160,000 applications to vote received in the final 24 hours of registrations.
Speaking on the BBC's Daily Politics programme, backbench MP Simon Danczuk said a third of the Â£3 members he had analysed from his Rochdale constituency had not voted Labour at the last election.
He said there was "clearly infiltration by some people in an attempt to get a far left candidate elected as  leader of the Labour party".
But ex-leader Ed Miliband's former director of communications Tom Baldwin told the programme: "I am astonished that so many people in the Labour Party have spent time whinging about the rules rather than doing what Jeremy Corbyn has and making the rules work for them."
Labour has said the contest will not be delayed, after fears some supporters had not received their ballot papers.
The party acknowledged that "a small number" of the 550,000-plus people with a say in the election had complained that they were still unable to cast their vote.
But a spokesman said all ballots had been sent out as planned.
Andy Burnham has urged Labour to "fight the Bullingdon Boys, not Tony Blair", as candidates chase last-minute votes in the leadership contest.