LA Dodgers co-owner Todd Boehly is understood to be the preferred bidder after being selected by Raine Group.
But there is a risk of talks folding and any deal being blocked as Chelsea are now demanding a change in the process.
Abramovich initially said he did not want his mega-money loan back.
But the Blues now argue that the debt “owed” to Abramovich through the club's parent company Fordstam should be sent as part of the proceeds of sale to Jersey-based Camberley International Investment.
Abramovich's name does not appear on any financial listings of the Channel Islands concern but it is believed he is linked to the company.
The United Kingdom Government believes the Russian would be banking a significant portion of the sale funds, despite his previous public statements insisting that was not a concern or interest.
But ministers have made it clear that they will not accept the Russian benefitting in any way from the process after he was placed on the UK sanctions list for his support of Vladimir Putin's invasion of Ukraine.
The idea that Abramovich or a related company will be able to walk away with £1.6bn - albeit money that he has previously loaned the club over the past 19 years - was described tonight as “a non-starter”.
It has been reiterated to Chelsea that the club's operating licence only runs until May 31 and that the club is now, genuinely, on borrowed time.
Unless a deal that is acceptable to the Government - and which ensures Abramovich does not receive the cash - goes through by that date, the possibility of the club being banned from the Premier League is a real one.
And it may even end up consigning 117 years of Stamford Bridge history to the scrapheap.