The Champions League winners have agreed to pay a transfer fee of around €80 million to sign Tchouameni, 22, after seeing off competition from European rivals to land the player.
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The midfielder will undergo his medical at the club on Tuesday, which will be followed by a news conference.
Tchouameni is seen by Madrid as a potential long-term replacement for Casemiro, 30, and will form part of a new midfield generation at the Bernabeu alongside Federico Valverde (23) and Eduardo Camavinga (19).
Despite being turned down by top target Kylian Mbappe -- who decided to sign a new three-year contract at PSG in a last-minute change of heart -- Madrid have been busy preparing for this summer's transfer window.
ESPN reported in January that Madrid were looking at options to rejuvenate their midfield, with Tchouameni the outstanding candidate. Madrid's chief scout Juni Calafat visited Monaco that month to lay the groundwork for a possible deal.
While Madrid had reached an agreement with the player, their deal with Monaco was held up by Spain's classification of the Principality of Monaco as a tax haven, meaning the Ligue 1 club would have to pay 24% tax on their profits from the transfer.
Los Blancos want to refresh their midfield and believe Tchouameni has everything "to become the defensive midfielder of the future."
Tchouameni made 35 league appearances for Monaco last season, scoring three goals as they finished third in the table behind PSG and Marseille.