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Chester chairman Stephen Vaughan told the Daily Post: "I have contacted the Hull chairman Adam Pearson and hopefully he will find our offer acceptable. This isn't the first time we have tried to bring Kevin back to Chester, hopefully this time our approach will be successful."
Ellison has a year remaining on his contract, and was determined to see it out after turning down the move to Scunthorpe. However, Vaughan is keen to get his man.
He said: "I am not put off by transfer fees, and I have shown in the past that if a manager wants a particular player I will go out and buy him."
Meanwhile, Bristol City have also made an approach for the left winger, but are not ready to firm up their interest.
Robins manager Gary Johnson told the Evening Post: "We did make an inquiry but that's as far as it's gone. We're not looking to take it any further at the moment."
Tranmere Rovers are also tracking the 27-year-old.