According to Israeli media, medics said the three dead were men in their 50s.
Jordan has sealed its own section of the border and said it was investigating an incident that occurred when a Jordanian vehicle was unloading goods into the West Bank from an Israeli-controlled section.
Also known as King Hussein Bridge, this crossing is located midway between Amman and Jerusalem and is the only official crossing linking the West Bank and Jordan.
It is also the only entrance to the West Bank without passing through Israel.
The Palestinian Health Ministry says more than 600 Palestinians have been killed in a surge of violence in the West Bank since Hamas attacked Israel on October 7, triggering a war in the Gaza Strip.
Last week, after a massive nine-day operation, Israeli forces withdrew from the city of Jenin and its refugee camp in the West Bank, a militant stronghold and home to some 60,000 civilians.
Israel says it is trying to stop deadly Palestinian attacks targeting Israelis in the West Bank and Israel and is taking action to combat terrorism.
