The 26-year-old Romero, who has won 39 caps with Argentina, joined the Genoa club from Dutch side Alkmaar in 2011.
He help them to win promotion from second-division in his first year and played 32 league matches last season.
At Monaco he will compete with current first choice keeper Danijel Subasic of Croatia, who was inconsistent last season.
Ligue 1 newcomers Monaco have splashed out around 150 million euros (127.93 million pounds) in transfers to bring in some top players including Colombia striker Radamel Falcao from Atletico Madrid, and fellow countryman James Rodriguez and Portuguese Joao Moutinho from Porto.
They host Montpellier on Sunday.
($1 = 0.7500 euros)
(Writing by Gregory Blachier in Paris; Editing by Pritha Sarkar)