Sure, it is not explicit, but implicit in the text; you asked for a "Biblical basis" which that could be. But combined with other passages we get a clearer picture of the practice:
Deuteronomy 7:1-5, in expressing the prohibition against intermarriage, G-d says "he [i.e., the non-Jewish male spouse] will cause your child to turn away from Me and they will worship the gods of others." No such concern is expressed about the child of a non-Jewish female spouse. From this, we infer that the child of a non-Jewish male spouse is Jewish (and can therefore be turned away from Judaism), but the child of a non-Jewish female spouse is not Jewish (and therefore turning away is not an issue).
Leviticus 24:10 speaks of the son of an Israelite woman and an Egyptian man as being "among the community of Israel"
(above from
Judaism 101: Who Is a Jew?)
Also, we should not discount the actual centuries of the practice in Judaism. As noted earlier in the thread, not all Jewish "denominations" practice matrilineal descent now, but most do and this is a recent development other than among the Karaits, which many Rabbinic Jews do not consider to be "Jewish".
But as the sages say "Torah can be interpreted in 77 ways", so perhaps that is not all there is to it.