“The Qassam Brigades are ready and well prepared to confront the aggression and defend their people,” Issam Daalees, a Hamas leader, said in a statement, referring to Hamas’s armed wing. “The enemy must understand that it cannot unilaterally impose the rules of confrontation and it must bear the consequences of its stupidity.”

In Sderot, a piece of a rocket penetrated the roof of a house and crashed into the living room.

Albert Hofi, the owner of the house, told an Israeli television reporter that moments before it was hit he had moved his disabled wife, Shula, to the safety of the basement. The rocket shard left a round hole in the ceiling and broke floor tiles, but the rest of the living room was intact.

“Unfortunately we have gotten used to the situation,” Mr. Hofi told the reporter, explaining his calm demeanor. A rocket alert sounded as they spoke, and Mr. Hofi and the television crew headed to the basement.

The week began with a rare sense of possible progress. A high-level delegation of Hamas officials, including some living in exile, convened in Gaza to discuss Egyptian and United Nations proposals to stabilize the cease-fire with Israel in return for an easing of the Israeli and Egyptian-imposed blockade of Gaza.

Mr. Netanyahu postponed a visit to Colombia and met with his security cabinet last Sunday for a strategic discussion about the situation.

Yet there were no signs that a broad deal was imminent. At the end of the hourslong cabinet meeting, Israel released a terse statement saying the military chief of staff, Lt. Gen. Gadi Eisenkot, had “updated the security cabinet on the situation regarding Gaza” and that the military was “prepared for any scenario.”

A senior Hamas official, Khalil al-Hayya, suggested earlier Wednesday that Hamas was open to a deal, but he accused Israel of violating the cease-fire agreement and added that Hamas would not allow Israel “to impose new equations or rules of engagement on the ground.”