Since September 2015, Israeli forces have killed more than 254 Palestinians, majority said by Israel to be attackers.
Some reports in the Israeli media also suggested the attackers were Israeli citizens of Palestinian origin.
According to the news site NRG, the mufti was called in to answer questions over his call to worshippers after the attack to ignore a temporary ban on prayer and public gathering at the Temple Mount compound and the Haram al Sharif mosque.
Israeli police on Friday detained Jerusalem's top Muslim cleric Sheikh Mohammad Hussein near Al-Aqsa mosque.
Since October 2015, more than 285 Palestinians have been killed by Israeli forces during - or immediately after - alleged attacks on Israelis or in clashes with Israeli security forces, according to Palestinian figures.
Abbas Zaki, of the central committee of al-Fatah, said Israel was morally responsible for the Jerusalem attack.
Israeli Prime Minister Benjain Netanyahu made it clear that Israel would take all necessary actions to maintain security on the Temple Mount, without changing the status quo, in a phone call Friday with Palestinian Authority Chairman Mahmoud Abbas.
Dozens of Palestinian residents waited in the sun to be allowed into the Old City.
Palestinian militant group Hamas has praised the shooting attack in Jerusalem that left two police officers dead on Thursday night as a sign of a continuing "Intifada", or uprising, against Israel.
"I have very little information about it, but it doesn't mean you should close the mosque for prayers", he told journalists at the Lions Gate entrance to the Old City, near the holy site.
Jordan has urged Israel to reopen the holy site. At least one had a knife and tried to stab Israelis, he said.
A number of people were wounded in the attack, police said.
The Old City, a hotly contested area, is in East Jerusalem, which Israel captured from Jordan in the 1967 war and then annexed in a move that was never internationally recognized.
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