Finance Minister Moshe Kahlon, leader of center-right Kulanu party and a major coalition partner, told Netanyahu that snap elections should be called as soon as possible in order to keep the economy on track.
Early elections have loomed since March, when Netanyahu's fragile coalition was sent into a tailspin after the head of the ultra-Orthodox United Torah Judaism (UTJ) party Yaakov Litzman threatened not to support the government's 2019 state budget if a draft bill to exempt religious Jews from military conscription was not approved.
The source close to Bennett told Reuters that an early election date will be decided on Sunday. Israeli law demands election campaigns need to be held 90 days from when the government is dissolved.
Sinwar said that he has sent a message to Israeli Prime Minister Benjamin Netanyahu and "his new defense minister" that "Gaza only prepares death for you and the rockets of resistance have become more in terms of numbers, accurate and destructive".
But Yisrael Hayom said Bennett was key to efforts to avoid an early election and could yet prove Netanyahu's political salvation.
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Netanyahu's coalition has 61 seats in the 120-member Knesset. The party said this was an "ultimatum" for it to stay in the government.
Education Minister Naftali Bennett speaks at an education conference in Ramat Gan on November 15, 2018. Without Bennett's Jewish Home, Netanyahu's coalition would lose its parliamentary majority.
Bennett has long criticized the Netanyahu government's reluctance to respond more forcefully to Gaza rocket attacks, and has advocated ground incursions into the Gaza Strip.
Hundreds of Israelis from Israeli communities surrounding the besieged Gaza Strip, participated in a protest in Tel Aviv, in central Israel, on Thursday, against "a weak reaction" on the part of the Israeli government after agreeing to a ceasefire with the Hamas movement.
Netanyahu said in his statement after the meeting that he had told Bennett of his intention to keep the defense portfolio "in the light of the critical challenges now facing the State of Israel". He warned the right-wing parties that rushing to a vote would be a historic mistake that could see the left take power.