At the time of Christ, there wasn’t one set canon that all Jews agreed upon. (If I recall correctly, that didn’t actually happen until several centuries after Jesus). So it’s not as though all the Jews at the time of Christ agreed upon the status of the deuterocanonical books.
The Hebrew Bible, in Jewish use, is divided into three sections: the Law, the Prophets, and the Writings. From what I’ve read, I believe the first two sections were already set in canonical form in the Herodian period, but the third not until some years later. That doesn’t mean, of course, that individual books in that section were held to be unsuitable in any way. The first book in the Writings section is Psalms, frequently quoted in the NT, even though, as far as anyone knows, it had not yet been formally incorporated into the Biblical canon in the time of Jesus.
The Law (
Torah): Genesis, Exodus, Leviticus, Numbers, Deuteronomy.
The Prophets (
Nevi’im): Joshua, Judges, 1 & 2 Samuel, 1 & 2 Kings, Isaiah, Jeremiah, Ezekiel, and the Twelve (the minor prophets from Hosea to Malachi)
The Writings (
Ketuvim): Psalms, Proverbs, Job, Song of Songs, Ruth, Lamentations, Ecclesiastes, Esther, Daniel, Ezra, Nehemiah, and at the very end, 1 & 2 Chronicles.