Israel hopes its violent standoff with Hamas over a captured soldier will eventually produce a new cease-fire with the Islamic militants, an Israeli Cabinet minister said Saturday, as Israeli troops exchanged fire with Palestinian gunmen and army bulldozers searched for militants' tunnels.

Until now, Israel had set only two goals for its military campaign in Gaza — to win the release of the soldier, Cpl. Gilad Shalit, and to halt Palestinian rocket attacks on Israel.

Cabinet minister Ofir Pines-Paz said Saturday that Israel wants to go beyond that.

"We have a great interest in changing the rules of the game," Pines-Paz, a member of the moderate Labor Party and of Israel's Security Cabinet, told Israel Radio. "If we reach a situation in which there are no kidnappings, no rockets, no tunnels, no raids into our territory, certainly Israel will have to reciprocate."

Hamas officials offered contradictory responses.

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Mushir al-Masri, a spokesman for the Palestinian ruling party, suggested rocket fire could end if Israel stops its offensive. However, the group said in a statement on its Web site that rockets "are the only available means for the Palestinian people to defend themselves in the face of the aggression and Zionist incursion into Gaza."

The latest round of fighting, which claimed the lives of 35 Palestinians and an Israeli soldier over the past three days, began two weeks ago with a cross-border raid in which Hamas-allied militants seized Shalit.

Troops initially entered southern Gaza where the soldier is being held. Hamas said Friday he is being treated well, and a senior Israeli defense official, Amos Gilad, said Saturday that Israel also believes the soldier is alive.

On Saturday morning, dozens of tanks drove toward Gaza City, taking up positions about 500 yards from the outlying neighborhoods of Shajaiyeh and Zeitun. The army said the forces were sent to the area to search for tunnels being dug by militants for possible attacks on soldiers.

The air force fired missiles at a group of militants gathered at the outskirts of Shajaiyeh. Two Hamas gunmen were killed in the area, hospital officials said. Also, a Palestinian died of wounds sustained in earlier fighting, bringing the three-day total to 35.

The majority of the Palestinians killed since Thursday were gunmen, but also included a number of civilians, including an 11-year-old boy.

At midday Saturday, about 250 Palestinians conducted a funeral procession in downtown Gaza City, carrying the bodies of two Palestinian militants who had been killed earlier in the day in fighting with Israeli forces.

Also Saturday, 65 U.S. citizens, many of Palestinian origin, left Gaza in a convoy escorted by U.S. consular officials. The visitors had asked to leave Gaza because of the fighting.

In northern Gaza, troops pulled back from the town of Beit Lahiya on Saturday, leaving a path of destruction. Tanks driving through narrow streets had shorn off outer walls of buildings, torn down electricity polls, carved up asphalt. Facades of buildings were marked by bullet holes from exchanges of fire.

Palestinian farmer Aref Sultan, 45, seized the brief lull to move his wife and seven children, ages 2-16, to a relative's home farther from the fighting. Sultan said he and his family had been pinned down in their Beit Lahiya home during the past day.

"We went through 24 hours of terror," he said as the family loaded food and clothing into a pickup truck. "Shots were coming from all directions ... and the tanks were approaching our house. The children, and especially my 2-year-old son Samir, were screaming all the time."

Israeli military commentators have said it would be difficult for Israel to extract the soldier in a military operation. However, Israel also does not want to be seen as cutting a deal with Hamas.

Under an Egyptian proposal, Hamas would release the soldier and halt rocket attacks on Israel, and in a later stage, Israel would free some Palestinian prisoners and halt its military offensive.

Israel's minister of public security, Avi Dichter, said Friday that Israel would consider freeing some prisoners if calm was restored. However, another Cabinet minister, Roni Bar-On, said Prime Minister Ehud Olmert opposes the release of prisoners in exchange for Shalit and would not negotiate a deal.

Pines-Paz said there was no contradiction between the two positions.

He said Olmert has embraced the Egyptian proposal, which does not call for negotiations with Hamas or freeing Shalit in a direct exchange for Palestinians held by Israel.

Pines-Paz said Egyptian President Hosni Mubarak has proposed that Shalit be freed without preconditions. "Afterwards, as part of a comprehensive agreement, without setting a timetable, it will be possible to release some prisoners, women, minors and so forth," he told Israel Radio."

The minister also said Israel hopes the fighting will yield a new cease-fire with Hamas. The militant group had largely observed an informal truce for more than a year, but the arrangement broke down over renewed fighting.