McGregor was first choice for the Turkish giants and played 26 times in the league last season, but the Scotland international is said to have become unsettled with life in Istanbul.
The 31-year-old joined Besiktas from Rangers - where he started his career - last season after he objected to his contract with the Gers being transferred to the newly-formed club, which started life at the bottom of the Scottish Football League in 2012/13.
"Negotiations with Allan are progressing well and there are no major obstacles to a deal being done," McGregor's agent Dave Baldwin said in The Sun.
"Allan is keen to play in the Premier League and very keen to play under Steve Bruce, so now we just have to push the talks forward."
McGregor has been capped 29 times for Scotland and was in manager Gordon Strachan's squad for last Friday's superb 1-0 World Cup qualifying win over Croatia in Zagreb.
Should the goalkeeper move to Hull, he will join up with compatriot George Boyd, who has penned a two-year deal with Steve Bruce's side after spending last season on loan from Peterborough.