The 46-year-old won three Russian titles during his seven-year stint at CSKA Moscow, while also leading Russia to Euro 2016 after taking over from former England boss Fabio Capello.
The Tigers, meanwhile, are looking for a new manager after Marco Silva left the KCOM Stadium to take the reins at Premier League side Watford.
According to Sky Sports News, Hull have narrowed the candidates down to a final shortlist and their month-long search for a new boss will be concluded before the weekend.
The report goes on to suggest that Slutsky, who has been mentored by Chelsea owner and fellow Russian Roman Abramovich, is the candidate most likely to be picked for the role.
Hull will play in the Championship next season after suffering relegation from the top flight in the 2016-17 campaign.