On Tuesday, an Israeli strike targeted and killed Taleb Abdallah, one of Hezbollah’s senior commanders, prompting the group to step up its attacks on Israel in retaliation. On Wednesday, it fired more than 200 rockets at Israel, according to the Israeli military, but they did minimal damage.

The Israeli military said on Thursday that its fighter jets had struck “Hezbollah military structures” overnight in Lebanese border villages.

Israeli officials have threatened stronger action against Hezbollah, and pressure to do so — from the political right and from displaced civilians — has been rising. But so far, both sides have stopped well short of full-blown war.

The United States, France and other mediators, warning of the danger of a regional war, have sought to advance a diplomatic settlement between Israel and Hezbollah that could restore calm on both sides of the border. But analysts say that the likelihood of an agreement is low as long as Israel’s campaign in Gaza persists.

On Thursday, residents of southern Gaza reported heavy bombardment by the Israeli military.

Saeed Lulu, who was sheltering in the southern region of Al-Mawasi — parts of which Israel has designated as a “humanitarian zone” for civilians — said he heard strikes between midnight and 6 a.m. The attacks, he said, appeared to hit the southwest edge of Al-Mawasi.

“We are very concerned,” Mr. Lulu, 37, said. “This is supposed to be a safe area, and we have no other place to go to if they attack here.”

Wafa, the Palestinian Authority’s official news agency, reported that Israel had intensified missile and artillery strikes on Al-Mawasi, where many Gazans have gone to seek refuge, heeding Israeli warnings to flee nearby Rafah, where hundreds of thousands had sought shelter earlier in the war.

The Israeli military denied the Wafa report, saying that it had not attacked the “humanitarian area” of ​​Al-Mawasi and that it was pressing ahead with operations in Rafah, where its soldiers were engaged in “face-to-face encounters” with Hamas militants.

Fighting in Rafah has raged on and off since early May, when Israeli soldiers began their advance into the heart of the city in what Israel has called an essential step in defeating Hamas’s remaining battalions and dismantling the group’s infrastructure.