Adding to Jason's answer:
In "The Fall of Arthur" (Tolkien's version of the Arthurian stories, unfinished but post-mortem published by his son with plenty of worthy commentary) Christopher Tolkien quotes some interesting notes by his father confirming that he at least at one time intended to bridge the gap between Arthur and the stories in Arda:
He [Lancelot] turns from her [Guinevere] and rides west.
The hermit by the seay shore tells him of Arthur's departure.
Lancelot gets a boat and sails west and never returns.
-- Ëarendel passage.
On the next page, Christopher quotes what (probably) is referred to as the 'Ëarendel passage':
O! wondrous night
when shining like the moon, with shrouds of pearl,
with sails of samite, and the silver stars
on her blue banner embroidered white
in the glittering gems, that galleon was thrust
on the shadowy seas under shades of night!
Ëarendel goeth on eader quest
to magic islands beyond the miles of the sea,
past the hills of Avalon and the halls of the moon,
the dragon's portals and the dark mountains
of the Bay of Faery on the borders of the world.
And later he quotes the second version of The Fall of Numénor:
But when Morgoth was thrust forth, the Gods held council.
The Elves were summoned to return into the West, and
such as obeyed dwelt again in Eresseëa, the Lonely Island,
which was renamed Avallon: for it is hard by Valinor.
The Fall of Arthur (ISBN 978-0-00-748994-7) pp.136
Christopher goes even farther analyzing this but I recommend buying the book. In my edition, the pages from 123-169 are all dedicated to 'The Unwritten Poem and its Relation to The Silmarillion' -- so plenty to read here.
Yes, Tolkien has at some point at least planned to merge the Arthurian stories into his. The link would've been the quoted 'Ëarendel passage' with Lancelot following his king to Avalon/Tol Eressëa (nobody knows whether he's found him and he never returns).
Incidentally Gawain's ship is called 'Wingelot' which is also the ship with which Ëarendel seeks for Valinor -- as Gawain dies in the Arthurian stories, in 'The Fall of Arthur' the former-friend of Gawain, Lancelot, apparently has the ship now. This however is more speculation and also contradictory to 'The Silmarillion' where is stated that 'Wingelot' was built by Cirdan for Ëarendel.
(bold and [in brackets] mine)