"I have instructed [law enforcement] to close Al-Jazeera's Jerusalem office because it promotes violence over Temple Mount", Netanyahu wrote on his Facebook page. "If that is not possible according to the current laws, I would work for the passing of necessary legislation at the Knesset to dismiss Al-Jazeera".
Al Jazeera was not immediately available for comment on Netanyahu's accusation, but the channel usually denies such an accusation.
Israeli authorities closed the Al-Aqsa Mosque that is venerated by Muslims and Jews - who call the site Temple Mount - and canceled weekly Friday prayers for the first time in almost five decades, following a deadly shootout July 14.
Mass waves of protest erupted following the imposition of heightened "security measures" at the entrance to the mosque compound, which is the third holiest Islamic site.
By threatening Al Jazeera, Netanyahu follows in the footsteps of the Saudi-led bloc of nations in its wide-ranging conflict with Qatar, which has accused Doha of supporting terrorism.
On Tuesday, Israel removed the metal detectors and planned on replacing them with less obtrusive surveillance.
Egypt and Jordan are now the only two Arab countries with peace treaties with Israel. "It's pure propaganda, of the worst variety, in the style of Nazi Germany or Soviet Russia", the Israeli Defense Minister Avigdor Liberman said, the Times of Israel reported.
A number of Arab countries have recently shut down their local al-Jazeera offices amid an ongoing spat with Qatar.