The Chelsea Supporters' Trust (CST) said it had held a meeting with representatives of Jim Ratcliffe, the British billionaire who tabled a last-ditch £4.25 billion ($5.3 billion) bid for the club last Friday.
Ratcliffe's offer came weeks after the deadline for interested parties to submit their bids to American bank Raine, who are overseeing the sale process, and has been "rejected out of hand" according to Tom Crotty, a director of his petrochemicals company Ineos.
Chelsea owner Roman Abramovich put the club up for sale on March 2 before being sanctioned the following week by the British government over Russia's invasion of Ukraine.
A special licence issued by the UK government allowing Chelsea to operate expires at the end of the month.
There are fears over the future of the club should a sale not be completed in the coming weeks with British culture secretary Nadine Dorries warning Chelsea are on "borrowed time."
"It is essential that the sale of Chelsea is completed swiftly and transparently," the CST said in a statement.
"We urge the current ownership, Raine and the UK Government to ensure this happens.
"Any delay to this process could risk the future of Chelsea - all parties must ensure the sustainability and future of the club is safeguarded."
The bid fronted by Los Angeles Dodgers co-owner Todd Boehly has received preferred bidder status from Raine.
Boehly has united with fellow Dodgers co-owner Mark Walter, British businessman Jonathan Goldstein, Swiss billionaire Hansjorg Wyss and US investment firm Clearlake Capital.
Under the terms of their licence, Chelsea are currently unable to offer new contracts to existing players or sign players from other clubs.
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