Consider the negation of "can music be considered literature?" The negation is: "music cannot be considered literature." From a strictly logical perspective, that would mean that there is no music that is also literature - there's not even one example of it. In other words, from a strictly logical perspective, all you have to do is find a single example of music that's also literature to prove that music can be literature.
Consider, by way of analogy, how you'd prove that there are unicorns, or that Leprichons exist. The simplest way to do so would be to find one; if you do find them, then clearly Leprichons exist.
I recall singing "The Sea" as an undergraduate, which was set to a text by William Dean Howells. Beethoven's "Ode to Joy" was similarly based on a Friedrich Schiller text.
There are thousands of songs based on religious texts as well. Notably, Handel's Messiah is taken very directly from Biblical texts, as are some of his other songs. Johann S. Bach also set many Biblical texts to music, such as Matthew 26 and 27. In contemporary music, there's a band called Sons of Korah whose songs are set (usually very literally) to the New International Version translation of various Psalms. I personally sang or chanted numerous Biblical texts as part of the weekly liturgy (in fact, typically at least 1 - 2 per week) in choir while I was in college.
My point in mentioning that is that those don't magically cease to be literature simply by virtue of having been set to music. Handel's Messiah is just as much literature as the underlying Biblical texts that it's based on.
My criteria, then: if the underlying text is literature, the music is literature, too. Note that I'm being a little cagy on the issue of the exact definition of literature, which I think is a separate issue - my point is simply "if the lyrics are literature, the song is literature too."
I think that this reasoning applies equally to things that were published as music first and were either published as literature on its own later (or have never been published as literature at all). For example, "Hotel California" and some other Eagles songs are very poetic, as are many U2 songs; I think you'd have a hard time arguing that those aren't literature, even though they've never actually been published separately in any collections of literature.
A more obvious case is Bob Dylan, some of whose works have been published independently of the music as literature (in poetry collections) and have been the subject of academic literary study. Even though they were songs before they were published as poetry, that doesn't change the fact that the song lyrics are, in fact, poetry.
You can actually read Bob Dylan's Nobel Prize in Literature lecture here, which includes some interesting thoughts on the question of whether his works are poetry. (I actually think that it would be worth creating a separate Q&A to discuss his lecture, but I haven't had time to write a question on it yet). He makes an interesting analogy: yeah, music was meant to be sung, but by the same token, Shakespeare was meant to be performed. If we want to argue that music isn't literature, we'd have to make the same argument about Shakespeare, Ibsen, Arthur Miller, and a host of other playwrites.
That being said, if we don't have a problem with plays being literature, why do we have a problem with music being literature, too?