The Tigers are keen to bring in a top-level successor to Marco Silva, who left last month following relegation to take charge of Watford.
Slutsky, 46, won three Russian League titles with CSKA Moscow and was also in charge of the national team at Euro 2016 when Russia crashed out at the group stages along with England.
He quit CSKA last December and has been living in England since the start of the year as a guest of Chelsea owner Roman Abramovich, Slutsky having spent the intervening months learning English with a view to working in either the Premier League or Championship.
City officials are believed to have met Slutsky last week and he has since returned to Russia. He is due back in this country at the start of next week.
Other candidates are under consideration by the Tigers, including Garry Monk following his recent departure from Leeds United. But Slutsky is believed to have impressed at interview and would represent a major coup by the Tigers, not least because there has been interest in the Russian from elsewhere in English football.
Hull are also understood to be closing in on the appointment of a director of football/head of recruitment, as part of a restructure of the set-up at the KCOM Stadium.
Silva’s departure along with all his coaching staff has brought an opportunity to totally overhaul football operations and move further away from the traditional English model of a manager being in charge of all facets at a club. An appointment could be made as early as next week.