The former international defender joins the Dutch club after leading Sparta Prague to successive league titles in the Czech Republic.
"We certainly did not rush into the search for a new coach," said Dennis te Kloese, technical director of Feyenoord, in a statement on Wednesday.
"It was a process that lasted many months, in which several colleagues besides myself were closely involved and many coaches were intensively followed to see who best fitted our desired profile.
"As a club, we want to build on the growth we have experienced and the successes we have achieved in recent years.
"After going through that process, Brian Priske turned out to be the logical and best choice for everyone involved.”
Priske also won the Danish championship with FC Midtjylland in 2020 and spent one season coaching at Royal Antwerp in Belgium.
As a player, he competed in the Belgian league with Racing Genk and Club Brugge and was also on the books of Portsmouth in England.
Feyenoord were runners-up in the league this year under Slot but won the cup. Slot, who guided them to the title in 2023, is replacing Jurgen Klopp at Liverpool.