Scunthorpe boss Brian Laws says he approached City for the player a month ago, and Peter Taylor expects he'll be getting frustrated if the deal isn't resolved soon.
He told the Hull Daily Mail: "Right now, it's dragging on a little bit and I imagine Brian is getting a little bit frustrated.
"They've offered £100,000 and I've accepted it so the rest is down to Kevin now."
If the transfer goes through, it will mean City get a full return on the £100,000 paid to Chester when Ellison moved to Hull in January 2005. He was signed as a replacement for the injured Stuart Elliott, and both players have been in and out of the side for the last year.
Liverpool-born Ellison has scored twice in 28 starts and 13 substitute appearances in his year and a half at Hull City.
If he goes, City fans might not have long to wait to see him again, as City take on the League Two side in a pre-season friendly at Glanford Park on 22nd July.