The Belgian giant's sporting director Herman von Holsbeeck has confirmed the £21m deal, with the youngster set to undertake a medical in the principality later this week.
Tielemans has been linked with a move to some of Europe's biggest clubs since April, including City.
But the 20-year old has instead decided to join the Ligue 1 champions, which according to von Holsbeeck, has come from Tielemans desire to play regular football.
"The transfer fee will be £21m plus bonuses involved," Holdsbeeck told Belgian newspaper Nieuwsblad.
"Medical tests have been scheduled for Tuesday or Wednesday. Monaco is the ideal club for Youri.
"Youri really wanted to go to a club where the coach would love to work with him.
"Last season, he said to me: 'I won't go to Manchester where I am just another number on a list'."