The Palestinian news website Maansaid that, in an interview conducted with the shooter yesterday, he had said he would turn himself in to Israeli authorities. Spokesman Mark Toner said there is "absolutely no justification for the taking of innocent lives".
A 39-year old Arab resident of Jerusalem shot and wounded several people in the city on Sunday and was then killed by Israeli police, a police spokeswoman said.
A woman of about 60 and a police officer of about 30 were fatally shot by the assailant, according to hospital officials.
Hamas said the suspect was a member of the organisation.
The assailant shot five Israelis, including the dead, before being stopped by police.
Forty-eight hours after a fatal shooting near Sheikh Jarrah in occupied East Jerusalem, the Israeli media is still prevented from revealing the identity of the assailant, 40-year-old Palestinian Misbah Abu Sbeih.
A few days earlier, a Jordanian citizen was shot after allegedly trying to stab Israeli Border Police officers near Damascus Gate in the Old City.
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The US State Department released a statement condemning "in the strongest possible terms the terrorist attack that took place today in Jerusalem".
The attacks had waned in recent months but surged, as if on cue, after the Muslim feast of Eid al-Adha and in the prelude to the Jewish holidays, which began a week ago.
Sunday's attack in Jerusalem was the deadliest on Israelis since June 8, when two Palestinians opened fire and killed four people at a popular Tel Aviv food market. Since then, they have killed 36 Israelis and two visiting Americans.
The vast majority of the attacks have been carried out by lone-wolf assailants, Israeli authorities say.
Israel's Minister of Public Security Gilad Erdan, who arrived at the scene, told the press that there was no forewarning for this attack.
Israel blames the violence on a Palestinian campaign of lies and incitement.
Palestinians say the violence is the result of frustration due to decades of occupation. "In my opinion, Facebook and other social media sites bear direct responsibility".