The Whites first made a bid for Cooper last month after he impressed against Leeds in a pre-season friendly.
An initial bid was rejected as was a second as Leeds seemed reluctant to improve on a £500,000 asking price.
Cellino first made his move after Nicola Salerno recommended him after just one game.
Cooper is reportedly a Leeds fan and was interested in the move, but United turned their attention elsewhere when they confirmed they would not be making a third approach.
Juventus defender Frederik Sorensen was expected to join Leeds last week but his move collapsed due to a disagreement between the Whites and his agent.
With talks now appearing dead in the water, Leeds seem to have rekindled their interest in Cooper with Dom Poleon also rumoured as part of the bid.
Journalist Alan Biggs first reported the interest in Cooper last month and he now believes Leeds are close to agreeing a fee.
From what I hear, a deal either agreed or very nearly agreed for Liam Cooper to join Leeds from Chesterfield. #lufc #cfc— Alan Biggs (@AlanBiggs1) August 12, 2014
Whether Poleon will move in the other direction remains to be seen but it does appear as if Leeds have finally realised that the defence at present is below the required standard.
Scott Wootton's showing against Millwall may well have spooked Cellino, but he still needs to get this deal over the line after a number of moves collapsed in the last few weeks.
In other transfer news, Leeds look close to a deal for Giuseppe Bellusci. The Catania man was linked with a move to Elland Road earlier this summer but United couldn't agree a contract.
The centre-back has apparently landed in England today ahead of a season-long loan move.
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