Connolly was born in 1945 in Govanhill, Glasgow. The identity of his father was never made public and his mother was a teenage waitress, Frances Connolly, who left him in a Glasgow hospital as an infant whilst he was possibly suffering from meningitis. Connolly was fostered at the age of two by Jim and Helen McManus of Blantyre and took their family name. Connolly was the half-brother of Mark McManus of Taggart fame.
In a radio interview, Connolly reported that singing was a large part of growing up since there was no television, and that he was regularly called upon to sing for family and friends. Connolly has credited the Everly Brothers as being his earliest musical influence. After inadvertently discovering his lineage he eventually reverted to the name Connolly.