Conti was born in Lovia to Sanmarinese parents with his father being Luigi Conti, a former player of Azzuri FC who went with the team when in moved to Lovia in 1977. However, he has also spent time living in San Marino and has chosen to represent that country in international football, gaining a number of caps with the San Marino national football team. Conti entered the youth academies of San Marino Calcio in 2007, before being signed by Azzuri FC in 2009. Alessandro's cousin Giulio Conti is a journalist in Brunant.