The 46-year-old former Russia and CSKA Moscow boss replaces Marco Silva, who left after City were relegated from the Premier League last season and has since joined Watford.
Slutsky told the club website: "I am delighted to become the head coach of Hull City and I am very much looking forward to the challenge of guiding the club back to the Premier League.
"I already know that this is an important club with a big and loyal fanbase
I've had a good feeling about it ever since meeting (owners) the Allam family and I can't wait to get started next week."
Slutsky guided CSKA to three Russian Premier League titles before resigning last December, and also led the national side to the finals of Euro 2016.
The club say he will arrive in Hull next week to officially start his new job, subject to international clearance.
Vice-chairman Ehab Allam said: "I am delighted to welcome Leonid as our new head coach.
"After a thorough recruitment process, Leonid's attention to detail, thoughts on playing style and his ambition made him the stand-out candidate.
"The club has made plans for pre-season training and friendlies, with some exciting fixtures to be announced next week, and squad planning with Leonid is already under way.
"Leonid has enjoyed success at both club and international level and I am excited about his plans for guiding us back towards the Premier League."
Source : PA