The defender has attracted attention from a number of clubs after an impressive 2008/09 campaign with the Tigers.
Hull manager Phil Brown admitted earlier on Friday that he had fielded enquiries from three or four Premier League teams.
He remained hopeful of keeping the 25-year-old at the KC Stadium but it now seems Sunderland are edging closer to bringing him in.
Black Cats boss Steve Bruce has already captured a couple of defenders this summer in Paulo Da Silva and John Mensah but he has made no secret of his admiration for Turner.
Bruce will now be looking to push through a deal before Tuesday's transfer deadline.