Israel’s security cabinet was preparing to meet on Tuesday afternoon, officials said, as the military pushed ahead with a new offensive in Gaza while the fate of a cease-fire proposal remained in doubt.

Amid outrage at home and abroad over the intensifying war, protests began shortly after dawn on Tuesday across Israel. The demonstrations were aimed at pressuring Prime Minister Benjamin Netanyahu to accept a cease-fire deal with Hamas that would release some of the hostages who have been held in Gaza for nearly two years.

“Advancing the plan to conquer Gaza while there is an agreement lying on the table for the prime minister’s signature is a stab in the heart of the families and the entire nation,” said Itzik Horn, father of Iair Horn, who was released in February, and of Eitan Horn, who is still held hostage. The brothers were captured in the Hamas-led attack on Israel on Oct. 7, 2023, that started the war.

“Join us today in our collective struggle because only the people will bring them home,” Mr. Horn said in a statement released by the Hostages and Missing Families Forum, which represents the relatives of some of the captives.

Three Israeli officials confirmed that the security cabinet meeting, which includes senior ministers and is chaired by Mr. Netanyahu, would take place on Tuesday. It would be the first such meeting since Hamas agreed to the cease-fire proposal last week. But it is not expected that the ministers will endorse the proposal, which has been described as a “partial deal” that would release some hostages immediately, allow more humanitarian aid into Gaza and provide a path to discussions to end the war.

Israeli officials are instead pushing ahead with a military offensive in Gaza City, and have signaled that they want to negotiate a comprehensive deal that would bring home all hostages at once and disarm Hamas.