The Hull Daily Mail says the Tigers' head of football operations Adam Pearson has held talks with the 57-year-old in the last fortnight and Hart appears to be in pole position to succeed Phil Brown at City.
Pearson is expected to talk with other targets in the coming days - with Cardiff boss Dave Jones also on his shortlist - before making a final decision by this weekend.
However, any move for Hart could be hijacked by Crystal Palace - the club he successfully guided to Championship survival on the final day of last season.
Hart's short-term deal at Selhurst Park expired last month but it is understood a new consortium has approached the former boss to return and again lead the financially-stricken club next year.
Having fostered strong emotional links with the Eagles in the final three months of last season, it would be an offer Hart would take seriously.