According to the security officer of the US embassy Tel Aviv branch, the woman was spotted roaming the area in a suspicious manner several times.
The Prime Minister's son has long been a controversial figure on social media. Last month, he received criticism for saying that Tel Aviv should be "cleared of minorities."
The email containing the death threats contained many various conspiracy theories.
The messages told the 12-year-old boy that he would be slaughtered.
The threats made were a response to Regev's comments on Yaron London's sexual harassment ordeal, when she called on him to resign from his position as broadcaster for Kan.
Gallagher, who has been accused of war crimes, will go to trial in less than three weeks.