Conti spent his entire career, aside from two year-long loan spells at Genoa in Serie B during the 1975–76 (during which he won the Serie B title) and the 1978–79 seasons, at Roma. He had initially been a member of the club's youth squad between 1972 and 1974, eventually making his senior club and Serie A debut on 10 February 1974, in a 0–0 home draw against Torino, during the 1973–74 Season, under Nils Liedholm, at the age of 19. As a youngster, he had been overlooked by scouts during trials with several professional clubs, including Roma manager Helenio Herrera, who felt that his physique would not allow him to succeed at the top level, despite his talent.