UNITED NATIONS — A major escalation in cross-border attacks around Gaza is increasing the likelihood of a decisive military confrontation between Israel and Hamas, which could include an extensive Israeli ground offensive to end Hamas’s control of Gaza, Israeli officials and politicians say.

The death of a student Wednesday morning at Sapir College near Gaza was met with ferocious Israeli airstrikes that by last night had killed at least 31 Palestinian Arabs, among them four children. But politicians and officials in Israel are clamoring for more decisive military action to end what an influential and hawkish lawmaker, Yuval Steinitz, is calling “this war of attrition” between Hamas and Israel.

Defense Minister Ehud Barak said during a closed-door meeting with aides yesterday in Jerusalem that Israel has begun preparing “the international community” — foreign officials involved in negotiations with the Palestinian Arabs, including a former British prime minister, Tony Blair — for a “large ground operation” in Gaza, several Israeli press outlets reported.

“If there is one more fatality, Israel may have to go in,” a Western diplomat who asked not to be named said. Since the killing Wednesday of Roni Yihye by a rocket from Gaza — the first such Israeli fatality in several months — a growing number of Israeli officials, opinion makers, and politicians have searched for ways to counter the increasing accuracy and range of the Palestinian Arab rockets.

“This is not an easy decision, but it may be necessary. We have to break the Hamas military power,” a former Israeli deputy defense minister, Ephraim Sneh, told The New York Sun.

Mr. Sneh, a member of the Labor Party, added, however, that a major ground offensive could be conducted only when an “exit strategy” is devised. “When can you do it? When you know who the keys can be handed to once you leave Gaza,” he said. This means closer cooperation with the Palestinian Authority to build a “credible” military force under its command, he said.

But according to Mr. Steinitz, a Likud Party member who has long advocated a ground offensive, “Israel has no choice but to go to a Defensive Shield 2 now in order to demolish this Iranian satellite in Gaza.” Mr. Steinitz was referring to a 2002 operation launched by Prime Minister Sharon in which the Israeli Defense Force temporarily re-entered major cities in the West Bank after a suicide bombing in Netanya.

Mr. Steinitz compared Hamas-controlled Gaza, which he described as a “mirror image” of the theological state of Iran, to Hezbollah-controlled southern Lebanon before the 2006 war there. But while rockets from southern Lebanon reached only rural areas in northern Israel, Gaza is close to Israel’s major urban areas. “Today it is Ashkelon, tomorrow Ashdod, and then Tel Aviv,” he said, referring to Israel’s coastal urban centers.

The 2006 war in Lebanon was considered an Israeli failure despite significant Hezbollah losses in equipment and personnel. Defensive Shield, however, was hailed in 2002 as a major military success that almost completely ended Palestinian Arab suicide attacks in Israeli cities. As then, “what we need to do in Gaza is get in for a few weeks, get the necessary intelligence and control mechanisms, and then pull out,” Mr. Steinitz said.

Yesterday the residents of Ashkelon, a major city seven miles north of Gaza, experienced for the first time what people in the smaller town of Sderot, across the border from Gaza, have lived through daily for the last few years. At least four rockets landed in downtown Ashkelon, injuring a 70-year-old woman, and a warning system known as “Red Dawn,” which alerts residents to imminent attacks, was installed across the city.

At the same time, explosions were heard across Gaza while Israeli drones, manned aircraft, and artillery pounded the area. Targets included a truck traveling near the major Gaza medical center of Shifa. Israeli military officials said the truck held explosives, but according to Palestinian Arab reports, it was delivering soda bottles.

“I’m concerned about the humanitarian conditions there and innocent people in Gaza who are being hurt,” Secretary of State Rice said after a meeting with Prime Minister Olmert in Tokyo. But, she added, “We have to remember that the Hamas activities there are responsible for what has happened in Gaza, the illegal coup that they led against the Palestinian Authority institutions.”