Ranieri resigned on Sunday in the aftermath of the Italian capital side's 4-3 loss at Genoa, despite having led 3-0, for Roma's fourth straight loss.
That reverse left the club lying eighth in the table and nine points off the last Champions League qualification position, which is held by their bitter city rivals Lazio.
Montella, 36, is a former Roma striker. He spent six months on loan at English Premier League side Fulham in 2007.
Known during his playing days as L'Aeroplanino (the little airplane for his small stature and trademark goal celebrations) he earned 20 caps for the national team, scoring three goals.
He was a key member of the Roma side along with Francesco Totti and Argentina's Gabriel Batistuta, then coached by current England boss Fabio Capello, which won their third and last league title in 2001 In his ten years with Roma between 1999 and 2009 he scored a total of 118 goals.
This is his first major managerial position, his only previous experience coming as coach of Roma's under-15 team.
Roma are in the process of being sold to a consortium headed by the wealthy Italian-American industrialist Thomas DiBenedetto.