The exchange of fire happened at the Bekaot checkpoint in the Jordan Valley on the second Friday of the Muslim fasting month of Ramadan, during which Israel has eased some restrictions on Palestinian movement from the West Bank to Jerusalem.
"A Palestinian vehicle reached the Bekaot checkpoint, the terrorist in the car fired at soldiers," who "responded and identified a hit," an army statement said.
The army said it did not know the extent of the gunman's injuries.
On Sunday, a Palestinian from the West Bank stabbed an Israeli policeman near Jerusalem's Old City before being shot. Both were seriously wounded. In response, Israel revoked entry permits to residents of the knife man's village.
On Tuesday, a rocket from the Gaza Strip hit southern Israel. In response, Israel revoked permits for 500 Gazans to enter Jerusalem for Ramadan prayers.