I don't know if Lucas ever formally tried to get the rights. He was probably canny enough not to even attempt it by that point. Besides, Saul Zaentz had already picked up the rights to LOTR; while the Silmarillion rights have never been sold, and likely won't be until after Christopher Tolkien is dead.
Having said that, there are a LOT of things even in the earliest versions of Willow that were directly inspired by Tolkien's work. In fact, some scenes were originally more similar to prototypes in The Silmarillion than they ultimately turned out to be.
For instance, the journey to the Elven Otherworld of Tir Asleen (as it originally was) was done via water, like the journey to Valinor. And as with Valinor, it is normally impossible for mortals to reach Tir Asleen--in this case because a fearsome sea monster guards the way.
There was also a scene immediately before that, where Willow and Madmartigan find a ruined human castle. Willow uses magic to resurrect twelve knights inside who have been petrified by Bavmorda, and Madmartigan asks them to set out with him on the impossible journey to Tir Asleen.
Impressed by his courage, the knights decide to make him their new King, and join his quest. However, other humans hiding out in the castle from Bavmorda's troops decline to go, only to show up and save the day heroically, Han Solo-style, later in the film.
This scene has obvious parallels in the stories of the mortal heroes Tuor and Beren in The Silmarillion. Tuor finds a suit of armor in the abandoned Elven castle of Nevrast, which ends up leading him on a quest to find the hidden Elf city of Gondolin. And Beren pleads with the King of Nargothrond, Finrod Felagund, for help in stealing a priceless Silmaril from Morgoth's crown. Finrod and ten Elf warriors join Beren (for a total of twelve), but the other Elves refuse to go out of fear.
Plus Sorsha originally being a half-Elf is a clear reference to Arwen.
On the other hand, there are also some clear Narnia references. Sorsha is pretty clearly a female version of Shasta, the hero of The Horse and His Boy, who was kidnapped at birth and raised as a poor peasant in a foreign country. The plot of the novel concerns Shasta's journey home and his discovery that he is really a prince.
Bavmorda's petrification of the twelve knights also has echoes of the White Witch Jadis in The Lion, The Witch and the Wardrobe.
There's even a bit of Robert E. Howard thrown in. A couple of ferocious blue-painted warriors, originally known as Picts and later renamed Pohas in typical Lucas fashion, make a cameo appearance in early scripts. And General (originally King) Kael was obviously named after Howard's King Kull, literary precursor to Conan.