The Hull Daily Mail says having had a £1.8m bid accepted by current club Besiktas, the matter is now down to talks between the player, his agent and the Tigers.
Dave Baldwin, who represents the 31-year-old former Rangers star, says he is keen to complete a move to the KC.
"Allan is keen to play in the Premier League and is very keen to play under Steve Bruce, so now we just have to keep pushing the talks forward," Baldwin told the Hull Daily Mail.
"I wouldn't say there are any major obstacles to a deal being done, but there are a few things Allan wants to make sure are right in the deal, and Hull City also have areas they want to make right for them.
"As with any deal, I can't put a timeframe on it, but we are making progress."