The 67-year-old Argentine has been out of a job since he was sacked by the Elland Road club in February 2022.
He held talks with both Bournemouth and Everton in recent months over their then-vacant managerial positions.
Two-time champions Uruguay failed to progress out of their group at the World Cup in Qatar.
Bielsa joined Leeds in 2018 and is still revered by the club’s fans for overseeing their return to the Premier League after a 16-year absence.
Before moving to the Premier League, he managed in France, Italy, Spain and Argentina, as well as taking charge of Chile for four years between 2007 and 2011.
Marcelo Broli has served as interim head coach since February. Diego Alonso, who was appointed Uruguay boss in 2021, resigned following the World Cup.
Uruguay are 16th in FIFA’s world rankings.