He was born in New York, but moved to Garvaghey in County Tyrone at the age of four.
In 1998, he was made Ireland's first professor of poetry and in his later years, he moved to France, where he was made a chevalier de la LÃ©gion d'Honneur.
The Irish Arts Council has described his death as a "true loss to Irish literature".
It is expected his remains will be brought from Nice, where he was living, to be buried in County Tyrone.
The Irish President Michael D Higgins has led tributes to him, describing his work as "immense".
"Familiar with the literature of other languages, he was a careful translator and source of encouragement to others," he said in a statement.
His wry, self-deprecating company, his humour, his openness to opposite opinions, will be missed by all of us who were privileged to be his friends - and so many were."
John Montague stands alongside Seamus Heaney and Michael Longley as one of the most important Northern Irish poets of his generation.
He also led a remarkable life, and could call Northern Ireland, the Republic of Ireland, the USA and France home.
Born in New York in 1929, he was sent to live with relatives in Tyrone as a young child. Educated in Armagh and Dublin, he returned to the USA to write and study, teaching at Berkeley university in California in the late 60s.
He later moved to Cork, but eventually settled in France.
He published numerous collections of poetry and prose. His 1972 collection, The Rough Field, whose poems focused on Northern Ireland, is regarded as a classic.
In an interview with BBC Radio Ulster in June, Montague said he first travelled to France in 1948 because of the literature he had been studying.
"I decided I needed to know France. I'd been reading people like Flaubert, Baudelaire and so I wanted to actually get there," he said.
"On my bicycle I went because in those days you had to do all the crossing, so on my bicycle I went through England, aimed towards France, and I reached it."
He described the country as "so beautiful in a different way from Ireland, no storm clouds - a lovely ease."
The leading Northern Ireland poet John Montague has died at the age of 87.