McIlroy scored 131 goals in 497 appearances for Burnley between 1950 and 1962, helping the club to the 1960 First Division title and reaching the European Cup quarter-finals the following season.
The former striker also scored 10 goals in 55 appearances for Northern Ireland and was a key part of the squad that reached the last eight of the 1958 World Cup.
Jimmy McIlroy: 1931-2018.— Burnley FC (@BurnleyOfficial) August 20, 2018
The club is deeply saddened to learn of the death of our greatest ever player.
A giant of a man.
The thoughts of everyone connected with the club are with his family and friends at this sad time.
Rest in peace, Jimmy. pic.twitter.com/pHDU7i1NHy
"A wonderfully skilful and creative player, McIlroy went on to become a mainstay of the side that didn't finish outside the top seven in the First Division between 1956 and 1963 - the year he left the club to join Stoke City in a move that devastated the club's supporters," read a statement from the club.
"His home remained Burnley, where he lived for the rest of his life. The Clarets have lost a true club great and a part of their history, and the club's thoughts are with his family and friends on this sad day."