Jack and his tart-tongued wife Vera (Liz Dawn) were among the longest-running characters on the soap, which is set in a working-class community in northwest England.
Tarmey, a former laborer, joined CORRIE in 1979 and left in 2010.
A statement from the family of the actor read: "The family of Bill Tarmey have confirmed that he sadly passed away this morning in Tenerife. They have respectfully asked the media for privacy as they grieve for a wonderful husband, father and brother."
And a statement from ITV soap Coronation Street read: "The cast, crew and production team at Coronation Street are devastated to hear of Bill's death. Our thoughts and prayers are with Bill's wife and soulmate Ali and their family at this very sad time."
Jack P. Shepherd, who plays David Platt, tweeted: "Just heard the news about Bills death tragic news for everyone #RIPBILL."
Nigel Pivaro, who played the Duckworths' son Terry, said: "I am devastated to learn of the passing of someone who became a huge part of both my professional and personal life, who I came to look on as a second father. He was a very wise and dignified man who taught me a great deal and whose counsel I came to value greatly. He was a lovely man who cared deeply for his family and friends. We are all rightly bereft at the loss we will feel knowing that the world will be a little poorer without Bill to lighten it up with his warm humor."
Michael Le Vell, who has played mechanic Kevin Webster in the show for years, said: "Bill met my dad briefly when I first joined Coronation Street. And when my dad passed away Bill became like a surrogate dad to me. He and Liz looked after me and got me through those early days and we've been friends ever since. My heart goes out to Ali and his family. We knew Bill was poorly, but this is still a very great shock. I am going to miss him very much."
Tarmey suffered from ailing health over the years. He had a heart attack in 1976, followed by a stroke in 1977, and underwent a quintuple heart bypass in 1986.
Tarmey is survived by his wife and by their son and daughter.