This is a very interesting subject. For years I believed in the Pauline authorship of Hebrews, but deeper study has led me to question that.
Barnabus is a candidate for authorship. He was a Levite, so he was well versed in the Torah and he was a Pharisee, so he was also well-versed in the Midrashim.
Having said that, there are some Pauline turns of phrase in Hebrews. And certainly the author had very deep knowledge of Jewish law and Temple practices. But there are also many words and uses of grammar that are not typical of Paul's writing and, in fact, never appear in any of his other 13 epistles. And a Levite such as Barnabas would certainly be well acquainted with all of the Temple practices.
Apollos is another possible author of Hebrews as he was well versed in the Scriptures. But in fact, if it were not Paul who wrote Hebrews, then I definitely would lean toward Barnabus.
However ... ultimately it doesn't matter the man whom God used, it was the Holy Spirit who moved upon the person and it is the Word of God, so it is not dependent upon the man who wrote it.
If anyone wants a closer look at specific reasons against Paul's authorship, there is a short but well-reasoned article at
Authorship of Hebrews: Why not Paul? | SHARPER IRON. I recommend giving it a read.