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.