‘Akallabêth’

We have no specific date for the composition of Akallabêth, though by definition it must have post-dated the Downfall it describes, which took place near the end of the Second Age. It seems most likely it was prepared early in the Third Age, probably in the North-kingdom (as suggested by the evident Elvish influences in the story, and the fact that a copy was apparently available to Bilbo at Rivendell).

A reference to the ultimate origins of Akallabêth is in a note included in the commentary for The Drowning of Anadûnê (in volume IX 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 of the Downfall of Númenor.