Connelly scored 105 goals in 265 games for the Clarets and was the club's top scorer as they won the Football League Championship in 1960.
The winger won 20 caps for England during his 17-year career, including one in England's 1966 World Cup finals campaign.
Connelly was part of the Burnley side that reached the FA Cup final in 1962 before moving to Manchester United two years later where he helped Matt Busby's team claim the First Division title in 1965.
Connelly left Manchester to ply his trade at Blackburn, where he spent four years before finishing his career with a three-year stint at Bury.
The flag at Turf Moor will fly at half-mast as a mark of respect for Connelly and Burnley's players will wear black arm bands during their championship game at Cardiff on Saturday.