“No one really wants to do it because we understand that the achievements are not going to be very impressive on the one hand,” said Giora Eiland, a retired Israeli general and former national security adviser, “and also Hamas is not interested in another cycle of violence because they suffer from many problems already and they do not want to bring their territory to another level of misery.”

Among the targets struck by Israel was a Hamas tunnel leading into Israeli territory, according to the military. Israel also said it had targeted a squad of militants who were trying to launch more rockets into southern Israel. One Palestinian militant was killed, according to Gaza health officials, and three Palestinians were injured in earlier airstrikes on militant targets in Rafah, in southern Gaza.

The overnight exchange came after seven Palestinians were killed on Friday by Israeli fire during a particularly stormy day of protests. Four of the dead had crossed the fence that separates Gaza from Israeli territory and tried to reach an army sniper’s post, and one was armed with a knife, according to Israeli forces at the scene.

In recent weeks, the Palestinians have also resumed flying incendiary balloons from Gaza, some rigged with small explosive devices and others designed to set fires in Israel.

The Israeli defense minister, Avigdor Lieberman, warned during a visit to the Gaza Division on Tuesday, “We have all come to the understanding that the situation as it is today cannot continue.”

Mr. Lieberman, who is known for taking a hard-line stance toward the Palestinians, said Israel had tried using peaceful means to reduce tensions in the area, which have risen since border protests began in late March, including cooperating with international mediation efforts to restore and stabilize the cease-fire that ended the war in 2014.