Ballo-Toure, 22, leaves the team in second place in France's top flight for a Monaco side who lie second from bottom.
Sources in the principality say Monaco have agreed to pay 11 million euros ($12.6 million) plus bonuses for the player who began his career at Paris Saint-Germain.
"Thierry Henry will help me progress as a player, that helped convince me to come," said Ballo-Toure in an interview clip posted on Monaco's Twitter account.
He has signed a contract until 2023 and is the club's third January recruit.
The 2017 French champions have already signed veteran Brazilian defender Naldo from Schalke and ex-Marseille and Roma midfielder William Vainqueur from Turkish side Antalyaspor.
Monaco, who travel to Marseille on Sunday, are also expected to complete the signing of Spain star Cesc Fabregas from Chelsea in the coming days.