"Running over of border policeman -- the terrorist was neutralised -- eliminated," a police message said, adding that the officer was "apparently moderately to severely injured".
Jerusalem's Haddasah hospital said in a statement that it was treating a casualty of the incident "a man of about 20... suffering head and other injuries, his condition is critical."
It was not immediately clear if it was referring to the policeman or another person, and police did not clarify.
On Monday, four Israelis were wounded in two stabbing incidents in Israel. One of the two Palestinian attackers was a teenager from Hebron, police said.
The same day a Palestinian tried to stab an Israeli soldier at a checkpoint in the northern West Bank and was shot dead.