The former chancellor was brought in to help the Todd Boehly-Clearlake consortium in their attempt to take over the club.
Robey Warshaw, the City advisory firm that Mr Osborne joined last year, has signed up as an adviser to the consortium fronted by Mr Boehly.
The former Conservative minister, who left frontline politics in 2016 to amass a portfolio of private sector jobs including the editorship of the Evening Standard, is a lifelong Blues fan and season ticket-holder.
If Mr Boehly succeeds in his bid, Lord Finkelstein will join the board.
Robey Warshaw is an adviser to the bid, which includes Mr Boehly, Clearlake Capital, Hansjorg Wyss and others.
The fate of the club is in the hands of the government after Roman Abramovich was disqualified from being a director.
Raine Group, the merchant bank handling the sale, is expected to recommend a preferred bidder to the government in the coming weeks.
The field of potential buyers for the English soccer club fell from four to three on Friday when a group of people withdrew.
The consortium issued a brief statement to say that it had not submitted a final offer due to certain issues that could not be addressed.
A source close to the bid said that Mr Griffin decided against committing to a binding offer.
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Boehly's bid is competing against one fronted by Sir Martin Broughton, the former British Airways andLiverpool chairman, and which includes funding from the Crystal Palace shareholders and private equity billionaires.
The True Blues Consortium, which counts former Blues captain John Terry among its founding members, has thrown its support behind Mr. Pagliuca.
More than 10,000 fans of the club have expressed an interest in owning shares worth more than 150 million dollars as part of the deal that will see Mr Abramovich replaced as the club's owner.
Sky News reported earlier this week that Clearlake Capital, a San Francisco-based investment firm, would own at least 50 per cent of the club if the offer succeeds.
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Boehly and his co-investors Mark Walter and Mr Wyss would own minority stakes in the club.
Jonathan Goldstein is a London-based property developer who works closely with Boehly.
The final bids were tabled on Thursday, which were expected to value the club at more than 2.5 billion dollars.
Raine has told the remaining bidders that they may have to wait for clearance from the league before presenting a preferred bidder.
Robey Warshaw declined to comment on Saturday, while Boehly's consortium was contacted for comment.
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