Roman Abramovich and the government are at odds over the terms of the takeover of the club.
Sky News understands that government officials are unsure if a special licence will be issued before the end of the month.
People close to the discussions say that the ministers have yet to receive assurances about the destination of the sale proceeds.
Without a deal by the deadlines, the club could be excluded from the competition next season or even face the threat of administration.
People close to the situation said that a licence could be issued before the end of the month, which would allow the club to start next season under a new owner.
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The group majority-funded by Clearlake Capital, a Californian investment firm, and spearheaded by the LA Dodgers part-owner Todd Boehly, struck a binding deal with the oligarch and his advisers earlier this month.
The sale would bring an end to the 19-year tenure of Mr. Abramovich as the owner of Chelsea, who lost on penalties in the FA Cup Final.
The Blues have been operating under a government licence since March, when Mr Abramovich was sanctioned, prompting a quick-fire auction which attracted billionaires from around the world.
The Department for Digital, Culture, Media and Sport (DCMS) is in talks with the club about a new licence to speed up the sale.
On Monday, officials said they were still concerned about the fate of the loan to the club.
The club's parent company, Fordstam, has a loan of 1.54 billion dollars to repay, and Mr. Abramovich doesn't currently have access to the funding to do so.
It is thought that the loan can be repaid on the sale of the club.
The crunch week is for negotiations.
Prior to being placed on the sanctions list in March, Mr Abramovich said he intended to write off the loan to the club and give the net proceeds from the sale to a new foundation to benefit the victims of the war in Ukraine.
Officials say they have not received enough assurances from the current owner.
The licence will not be issued without those.
This would be a crunch week for the negotiations.
The former chancellor may try to smooth the path to completion after the renewed risk to the sale.
The Boehly-Clearlake consortium is being advised by Robey Warshaw, the firm where Mr Osborne now works.
The new owners have agreed to pay 2.5 billion dollars to acquire Mr. Abramovich's shares, and to invest 1.75 billion dollars in its stadium, academy, and women's team.
Sky News reported recently that the takeover would prevent Mr Boehly and his fellow investors from paying dividends or taking management fees for a decade.
The package of anti-Glazer clauses was designed to avoid the controversies which have plagued Manchester United since the Glazers took over.
The new owners will not be allowed to sell any shares in the club for ten years, as well as agreeing to strict limits on the amount of debt they can take on.
The takeover of Manchester United by the Glazer family saddled the club with expensive debt known as payment-in-kind notes and provided a focal point for fan protests.
A decade ago, Manchester United was floated on the New York Stock Exchange and the Glazers made hundreds of millions of dollars from dividends and the sale of shares.
The takeover must be approved by the premier league.
The takeover is subject to approval from the premier league and a special licence from the government.
People close to the deal have warned that it is not certain that it will happen.
Mr Abramovich is said to be determined to donate at least 2.5 billion to a new foundation that will benefit war victims, with a demand in the last stages of the auction that bids increase by at least 500m.
The Clearlake-Boehly bid was opposed by a group headed by Boston Celtics part-owner Steve Pagliuca and Larry Tanenbaum, the NBA chairman and Toronto Maple Leafs owner.
Ineos Group tycoon Sir Jim Ratcliffe made a late entry into the process, but his approach was dismissed by the advisers.
Uncertainty over the club's ownership is being blamed for the departure of key players, including Antonio Rudiger, the German centre-half.
The club has won 19 major trophies under Mr. Abramovich, making it one of the top sides in Europe.
The DCMS didn't comment.