His widow Lady Gambon and son Fergus said their "beloved husband and father" died peacefully in hospital with his family by his side, following a bout of pneumonia.
Sir Michael's family had moved to London when he was a child but he made his very first stage performance in Ireland, in a production of Othello in Dublin in 1962.
Dame Eileen Atkins, a longstanding friend of Sir Michael, told BBC Radio 4's The World at One he was "a great actor, but he always pretended he didn't take it very seriously" and that he had amazing stage presence.
He was nominated for Emmy awards for his role as Mr Woodhouse in an adaptation of Jane Austen's Emma in 2010, and for playing President Lyndon B Johnson in Path to War in 2002.
The actor, known as "The Great Gambon" in acting circles, had last appeared on stage in 2012 in a London production of Samuel Beckett's play All That Fall.
Taoiseach (Irish prime minister) Leo Varadkar paid tribute, saying: "A great actor.
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