Monk was recently appointed as Steve Evans' successor as head coach at Elland Road, agreeing a one-year rolling contract.
The former Swansea City boss made bringing his assistant at the Liberty Stadium, Clotet, a priority and according to the Yorkshire Evening Post, the deal is done.
Leeds and Clotet have agreed terms and although there has yet been no official announcement, he will be taking his place in the dugout at Elland Road alongside Monk.
The 39-year-old Spaniard joined Swansea in 2013 as academy boss and was promoted to the role of Monk's assistant in May the following year.
Clotet did have the opportunity to leave Swansea to take over at Brentford, but chose to remain at Swansea with Monk.
When Monk was sacked in December last year, Clotet also left the Liberty Stadium.