There is no consensus on the question as you've stated it. However, there is broadly consensus that "Homer" is more important as cultural heritage label, an appellation d'origine controlée, than as a historical individual
Some people (let's call them A) believe in a series of redactions that transformed the poems over time but which left detectable traces of the editing process. Others (U) believe in the idea of a monumental poet (though not usually the same person for both epics), and reject the traces seen by people in A as either poetic licence or just ignore them. Others again (N) believe that a monumental poet (or one for each epic) drew on pre-existing poetic traditions for his work, which people in A have mistaken for evidence of redaction. And still others (O) believe in the gradual crystallisation over time of an evolving tradition of rhythmic formulas, stylistic devices, set-piece scenes, and plot forms, which left a hodge-podge of strata of different elements that happen to look like the traces cited by people in A.
From a certain point of view there is an a priori presumption in favour of A because it's based on how the transmission process of pretty much every other work from antiquity works: an author writes a work, it gets disseminated by an error-ridden transmission process, potentially redacted, and then it is the job of the modern editor to excise the errors and restore the text to its pristine condition. The problem with that is that the criteria available to the A school of thought are nearly all completely and irreducibly subjective, barring a few genuine internal inconsistencies for which both N and O have alternate explanations: it's very easy to satirise A and the ludicrous confidence exuded by its adherents.
U developed pretty much as a resistence to A, partly out of a 19th-century desire to celebrate the notion of a Romantic genius poet (Homer), partly because some people have better things to do with their time than worry about why a textual source looks the way it does. Needless to say, that's not very robust either.
Hardly anyone belongs to the A group any more, and not many to the U group. Both N and O have more explanatory power, though they both have their weaknesses too.
Once upon a time it was a debate between A and U, but things haven't looked like that for decades. There's still debate, but on quite different topics. Every now and then someone will try to revive the debate in terms like "my school of thought is better than yours!! nya nya nya", but not many people buy into it. The latest iteration was a pair of books by M. L. West, the editor of the Teubner edition of the Iliad, called The Making of the Iliad (2011) and The Making of the Odyssey (2014), where he tried to combine A and U by casting the composition process as a single genius poet modifying his own work in a series of phases, while dismissing N off-hand and openly insulting people in O. One review describes it as "fascinating, meticulous, eye-opening, jaw-dropping"; in another, West "licenses himself to ignore virtually all actual research on the Iliad". It's an ugly mess and no one comes out of it smelling good.
Meanwhile, current scholarship continues devoting some attention to N and O -- but not obsessively, as was the case with A and U. N is intrinsically likely but plagued by a very frail, untestable methodology; O has a tremendous amount of explanatory power but flies in the face of most of the actual known data. So, no consensus.
But what there is consensus on is perhaps just as important. The idea of "Homer" is now best understood as a poetic persona, rather than a historical individual. In a very real sense it doesn't matter whether "Homer", or someone else of the same name, "wrote" the poems: what matters is that the name was used as a label for a particular cultural heritage. It was an extremely typical thing for poets of the Archaic period to serve as personas that any poet could adopt. The most glaring cases are wholly mythical or fictional poets like Orpheus, Musaeus, Linus, and Abaris: we can be confident that these figures never existed, but there were most certainly real poems that purported to be by them (though the surviving fragments are very fragmentary). The next level are those poets who had a much stronger identity in later tradition, who may have been real individuals, but whether they were real or not doesn't really matter: Homer, Hesiod, Terpander, Sappho, Archilochus, Solon. It didn't matter if Homer was real: either way, it wouldn't stop the Homeridai of Chios from laying claim to the poetic heritage of "Homer", and it wouldn't stop Cynaethus, in the late sixth century, from adopting the persona of "Homer" in the Hymn to Apollo where he claims to be "a blind man (who) lives on rocky Chios, whose songs will be supreme forever". Like I said at the start, these early poets are more appellations d'origine controlée than anything else.
Edit. For reference, in case anyone wants to delve into the scholarship, A, U, N, and O stand for "analyst", "unitarian", "neoanalyst", and "oralist". They're broad categories, not dogmas.