The Algerian, 31, joined the Foxes from Sporting Lisbon on a five-year deal in 2016 for £29m, a then club record.
He has also had loan spells with Fenerbahce in Turkey and at Premier League rivals Newcastle United.
Slimani, who has scored 13 goals in 46 Leicester appearances, could make his Monaco debut at home to Nimes in Ligue 1 on Sunday.
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— AS Monaco EN (@AS_Monaco_EN) August 21, 2019