A reference to the ultimate origins of Akallabêth is in a note included in the commentary for The Drowning of Anadûnê (in volume 9 of The History of Middle-earth). Here, Tolkien suggests that it derived from a 'Mixed Dúnedanic Tradition, by which he seems to mean that it developed from a combination of Elvish and Númenórean accounts.
In reality, the ultimate origins of the tale lay in a discussion between Tolkien and C.S. Lewis. Lewis' space-travel novels (starting with Out of the Silent Planet) ultimately arose from this, and it was intended that Tolkien would create a story of time-travel at the same time. That story, The Lost Road, was never completed, but some of its fundamental features - including the character of Elendil and the downfallen island kingdom - went on to evolve into the Tale the Downfall of Númenor.