The 40-year-old has been the permanent Barnsley manager since June 2016, but held the position twice before that on a caretaker basis.
The Championship club parted company with Thomas Christiansen on Sunday following a seven-match winless run - the ninth manager to leave Elland Road since 2013.
The Whites are now 10th in the Championship, seven points off the play-offs but last season's surprise play-off charge under Garry Monk was the first time in six years they had finished in the top half of the division.More news: Rose McGowan cancels appearances after public spat
Chris Hall, secretary of the Leeds United Supporters Club, told Sportsmail: 'It's another manager gone (eight in four years) but there is a lot of good going on at the club now that Radrizzani has taken control from Massimo Cellino.
Former Leeds boss Christiansen had a win ratio of 42.9 per cent at the Yorkshire club, winning 15 of his 35 games in the dugout.
He guided the Whites to promotion from League One and also secured a 7th place finish in the Championship during his tenure.
Christiansen, a former Barcelona and Spain striker of Danish descent, made his name in Cyprus and seemed to immediately settle in England, with Leeds topping the table in early September. It will be their eighth managerial appointment in four years.