Who or what sent the prophetic dreams to Faramir and Boromir?

Question

In The Fellowship of the Ring both Faramir and Boromir have prophetic dreams urging them to seek Imladris (Rivendell) :

[O]n the eve of the sudden assault a dream came to my brother in a troubled sleep; and afterwards a like dream came oft to him again, and once to me.

'In that dream I thought the eastern sky grew dark, but out of the west a light shone forth, and from that light a voice called out, saying:

Seek for the Sword that was broken;
In Imladris it dwells.
There shall be taken counsels
Stronger than Morgul-spells.
There shall be shown a token
That Doom is near at hand.
For Isildur's Bane shall waken,
And the Halfling forth shall stand.

Who or what is the genesis of this message?

Note : Tolkien has stated that some things in his Legendarium are a mystery, even to him. This might be one of those cases.

Answer 1

Tolkien never clarified the point

Although I should be fair here; the first draft of Boromir's dream is lost, having been written on a page that was later misplaced, as Christopher Tolkien notes:

The verse (if it was already a verse) that brought Boromir to Rivendell is lost in its earliest form with the lost page

^{History of Middle-earth VII The Treason of Isengard Chapter VI: "The Council of Elrond (1)"}

None of the subsequent drafts, nor any of Christopher Tolkien's notes, discuss the origin of the dream. Nor have I found anything in Letters to clarify his intentions here.

It's most commonly believed that the visions were sent either by Eru Ilúvatar or by the Valar; this is the opinion held by Thomas Shippey, a prominent Tolkien scholar, who wrote in The Road to Middle-earth (as reported in Reader's Companion):

In Middle-earth, one may say, Providence or the Valar sent the dream that took Boromir to Rivendell. But they sent it first and most often to Faramir, who would no doubt have been a better choice.

^{The Lord of the Rings: A Reader's Companion Book II Chapter 2: "The Council of Elrond"}