Immortality cannot be "given" to someone else. Not even Manwe, King of Arda, can give or take immortality; only Eru himself has the ability to change the fundamental kind (and he only did so twice - Luthien and Tuor). Thus, Frodo and company were to die in Aman:
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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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I've heard somewhere (and this is very vague, and probably unsubstantiated) that Bilbo was able to go, because Thranduil foresook going into the west,
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Tolkien never said anything like this.