Brazilian-born Motta played for Paris Saint-Germain from 2012 to 2018, with his only previous coaching experience with PSG's U19 team, which he managed until July 2019.
The 37-year-old takes over a Genoa side that has won just one game out of eight played this season against Fiorentina on September 1.
They are second from bottom of the Serie A table with 20 goals conceded after Sunday's 5-1 defeat to Parma.
Motta oversaw his first training session on Tuesday and will be officially presented at the north-western side on Wednesday.
The former Barcelona and Atletico Madrid midfielder played for one season with Genoa, before moving to Inter Milan in 2009, winning the treble the following season under Jose Mourinho.
He won five Ligue 1 titles at Paris Saint-Germain, and two La Liga crowns with Barcelona.
Motta earned 30 caps playing for Italy, reaching the finals of Euro 2012.
Genoa escaped relegation from Serie A last season on the final day under Cesare Prandelli, who was replaced by Andreazzoli in June.
Motta's first game in charge will be on Saturday against Mario Balotelli's Brescia, before travelling to Turin to play defending champions Juventus next week.