The Ligue 1 giants were in the market for a new left-back following the departure of Aly Cissokho to Valencia last week, and they have now snapped up the Argentina international as his replacement.
Monzon came through the ranks of the Boca Juniors youth academy and also wore the jersey of Betis before joining Nice in the summer of 2011.
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Meanwhile, former France Under-21 international Jeremy Pied has moved in the opposite direction for €3m.
The attacker was no longer needed at Lyon after failing to secure regular first-team action at his boyhood club, and has inked a deal until June 2016 with Nice.