Part of the source in question might be letter 246 (drafts, dated 1963, to a Mrs Elgar), where Tolkien notes that both Frodo and Bilbo still bore a mark of the One, and:
Quote:
(...) 'Alas! there are some wounds that cannot be wholly cured', said Gandalf (III 268) -- not in Middle-earth. Frodo was sent or allowed to pass over Sea to heal him -- if that could be done, before he died. (...) So he went both to a purgatory and to a reward, for a while: a period of reflection and peace and a gaining of a truer understanding of his position in littleness and in greatness, spent still in Time amid the natural beauty of 'Arda Unmarred', the earth unspoiled by evil.' (...)
JRRT, Letters
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Full context is best. Much later in 1971, JRRT wrote to Roger Lancelyn Green (in part):
Quote:
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'As for Frodo or other mortals, they could only dwell in Aman for a limited time -- whether brief or long. The Valar had neither the power nor the right to confer 'immortality' upon them. Their sojourn was a 'purgatory', but one of peace and healing and they would eventually pass away (die at their own desire and of free will) to destinations of which the Elves knew nothing.'
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Anyway, I think letter 246 is
maybe part of the source of what you read online.