The farmer's son who went on to become a world renowned Nobel laureate died in hospital in Dublin aged 74.
Mr Clinton and his wife Hillary said they were saddened to learn of the death of their "friend".
"Both his stunning work and his life were a gift to the world. His mind, heart, and his uniquely Irish gift for language made him our finest poet of the rhythms of ordinary lives and a powerful voice for peace. And he was a good and true friend," the Clintons said.
"We loved him and we will miss him. More than a brilliant artist, Seamus was, from the first day we met him, a joy to be with and a warm and caring friend - in short, a true son of Northern Ireland. His wonderful work, like that of his fellow Irish Nobel Prize winners Shaw, Yeats, and Beckett, will be a lasting gift for all the world."
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