The Champions League holders fought off competition from the likes of PSG and Liverpool to sign the Frenchman for a fee that could rise to around €100m.
The 22-year-old will be presented to the media next Tuesday after undergoing his medical.
He becomes the second summer signing at the Santiago Bernabeu after centre-back Antonio Rudiger, who has joined the La Liga champions from Chelsea on a free transfer.
Tchouameni has made 50 appearances in all competitions for Monaco last season.
He will join the compatriot Karim Benzema, Eduardo Camavinga and Ferland Mendy at the Bernabeu.