It's unclear whether the Beatles actually tried to acquire the rights, but at the time they announced their plans Tolkien's publisher was already in the middle of negotiating a contract with United Artists, and they would not have been available. Also, Tolkien hated the Beatles, especially John Lennon
Allen and Unwin had already began negotiating with United Artists in 1967.
Already by mid-1967 Allen & Unwin began to negotiate with United Artists, who wished to purchase film rights to both The Lord of the Rings and The Hobbit (though their interest was chiefly in the former work). A deal was struck at last in 1969.
J.R.R. Tolkien Companion and Guide - "Adaptations"
However the Beatles seem to have announced their plans to adapt The Lord of the Rings without ever even entering into a conversation with either JRR Tolkien or his publisher. It's unclear if they ever tried doing so, but by 1968 they still hadn't and at that the point the rights wouldn't have been available any more.
Tolkien heard about their "plans" the same way that the general public did and he was not happy, as he seems to have really hated the Beatles. His secretary Joy Hill, wrote a memo to his publisher about Tolkien's opinion on it.
At the end of 1968 there were reports that the Beatles might be involved with a proposed film of The Lord of the Rings. On 7 January 1969 Joy Hill wrote a memo to Rayner Unwin after a visit to Tolkien: ‘Professor Tolkien is getting more and more furious about this … because it seems that the Beatles are announcing plans in connection with the film. … He is livid that the Beatles have done this and loathes them anyway. … Particularly … he seem to have a thing against John Lennon’ (Tolkien–George Allen & Unwin archive, HarperCollins).
J.R.R. Tolkien Companion and Guide - "Music"
Two weeks later, his publisher, Rayner Unwin wrote back to Tolkien, expressing his own doubts about the news as he seems to not have heard anything about it either.
Rayner Unwin writes to Tolkien. He doubts that the Beatles are really involved with plans for a film of The Lord of the Rings; ‘however this may be, I think it is one of the wearisome things that we shall all of us, doubtless be subjected to as a price of the film contract once it is made! This is part of the show business world. I am perfectly certain that we shall both of us dislike intensely every manifestation of what is done to The Lord of the Rings.’ But he reminds Tolkien of their agreement: that if a film brings cash, they will waive any kudos. He points out that whatever a film is like, ‘the book remains inviolable and that is the main thing. What they do with the property in other media will, I regret to say, be entirely their responsibility from an aesthetic point of view, will only vary in degrees between bad at best and execrable at the worst’ (Tolkien–George Allen & Unwin archive, HarperCollins).
J.R.R. Tolkien Companion and Guide - Chronology "20 January 1969"
I'm unaware of the source, but according to Kristin Thompson, the Beatles did enter negotiations with United Artists, while UA was still in negotiations to acquire the rights but Tolkien threatened to halt the negotiations if the Beatles were going to be involved.
the Beatles conceived a desire to star in an adaptation of The Lord of the Rings. Their production company, Apple Films, already had a relationship with UA, which had distributed the group’s first two films, A Hard Day’s Night (1964) and Help! (1965) in the United States. Some negotiations apparently occurred, but Tolkien threatened to withdraw from the ongoing talks with UA, and the Beatles’ hopes were squelched.
A Companion to J.R.R. Tolkien - "Film Adaptations: Theatrical and Television Versions"