There is a passage in the Christian books that says that the Tanakh predicts that a prophet will come from the town of Nazareth. Does it actually say that anywhere? I want to know what, if anything, their text is based on:
"There he made his home in a town called Nazareth, so that what had been spoken through the prophets might be fulfilled, “He will be called a Nazarene." (Matthew 2:23)
The answer is that there is no prophecy in the entire Tanach which foretells that a prophet should come from Nazareth, or that the Messiah should be called a "Nazarene". Christians throughout the ages have tried reconciling the fact that Matthew's Gospel claims that there is a reference in the prophets to this, but there is not. Even Christian Bibles do not print a reference to this "claim". The simple answer is that the reference cannot be found at all within the Jewish Scriptures.
Therefore, it is not a requirement in Judaism that the Messiah come from Nazareth, nor that he be called a "Nazarene". (Which I'm not sure what the Christian Bible means by this.)
Keep in mind that questions such as this one in the future might not be within scope of the site.
As mentioned in heshy's answer, the Ramban, whom you've named yourself after on this site, was famous for his debates with the Christian authorities at the time, defending Judaism and explaining why Yeshu cannot be the Messiah. Read more about the Ramban's most famous dispute here.
Many christians have tried to prove thus over the centuries, and have debated the topic countless times. You very appropriately called yourself "Ramban", as there are published many debates between the Ramban and proponents of christianity on this subject. The position of all rabbis is that there is no mention of this particular prophet.
however there is this verse:
אַךְ הַנָּבִיא אֲשֶׁר יָזִיד לְדַבֵּר דָּבָר בִּשְׁמִי, אֵת אֲשֶׁר לֹא-צִוִּיתִיו לְדַבֵּר, וַאֲשֶׁר יְדַבֵּר, בְּשֵׁם אֱלֹהִים אֲחֵרִים--וּמֵת, הַנָּבִיא הַהוּא.
-Deuteronomy 18-20
The Chizuk Emunah chelek 2:6 uses this verse from Matthew as one of his many proofs to show the fallacies of the NT. He explains that there cannot be found 1 verse in the prophets which mention such an idea of the Messaih being called a Nazarean.
