“The people here in Gaza have had enough, and honestly they are pushing Hamas for a cease-fire,” said Mukhaimer Abu Saada, a political scientist at Gaza’s Al Azhar University. “People are mad at Israel, but they are also mad at Hamas — enough is enough. The situation that is existing now is just a lot of destruction, a lot of sadness, and people want to breathe. As long as Israel is not killing Palestinians, it’s like, O.K.”

There was still the possibility that Hamas’s rejection of the cease-fire would not be accompanied by a spray of rockets, a calm that Israel would presumably reciprocate, allowing a lull to continue despite a face-saving rejection.

Significant gaps remain on the parameters of a durable truce. Hamas wants the release of high-profile prisoners along with open border crossings into Gaza and a lifting of Israel’s restrictions on fishing, farming and trade. Israel wants international guarantees that Gaza will be demilitarized, including a monitoring system to ensure that imported supplies would not be used to rebuild tunnels.

The longer the temporary truces last, the more tunnels Israel can destroy, and the easier it would be for Prime Minister Benjamin Netanyahu to sell a cease-fire to his cabinet and constituents. The military said its troops found four new tunnel shafts during Saturday’s 12-hour pause; since the ground operation began July 17, a spokesman said, 31 tunnels have been unearthed and 15 destroyed.

In these cease-fires, “We are allowed to destroy the tunnels,” said Amos Yadlin, a former Israeli chief of military intelligence. “This was the mission anyway, so why not do it with no fire around?

“The dilemma for the Israelis will be, when we will end this mission? Is that enough, or do we have to go to another stage? Do we have to scale up to increase the pressure on Hamas?” he said. “But this is not today and not tomorrow.”