CAIRO -- Israel and the Islamist group Hamas inched closer Tuesday to agreeing to an Egyptian-negotiated cease-fire that would end a weeklong Israeli offensive in the Gaza Strip that’s killed at least 116 Palestinians and threatened to devolve into all-out war. Both sides suggested that the announcement of the cease-fire could come as soon as Wednesday.

With a cessation of hostilities possibly just hours away, both sides stepped up their attacks late Tuesday, with artillery fire so intense near the Israeli town of Erez on the Gaza border that computers were shaken from tables by the vibrations from outgoing shells.

“Each side will keep shooting until the last possible moment,” said one Israeli military official currently based along the Gaza border, who agreed to speak only on the condition of anonymity because he wasn’t authorized to talk to a reporter. “We each have to show that we didn’t retreat first.”

Meanwhile, Gaza militants launched scores of rockets, including one that struck the southern Tel Aviv suburb of Rishon Lezion. No injuries were reported.

As the final details of the cease-fire apparently were being refined, both sides claimed victory. Israel said the weeklong assault on Gaza had improved security for thousands of Israelis who’ve been targeted by Hamas-fired rockets. Hamas said the fighting had given it greater legitimacy among the Arab nations.

The United States, which was uncharacteristically on the sidelines, appeared to have successfully played a limited but key role in finalizing the deal.

U.S. Secretary of State Hillary Clinton, who rushed to Jerusalem from Phenom Penh, Cambodia, where she and President Barack Obama were attending a summit of Asian leaders, met for two hours with Israeli Prime Minister Benjamin Netanyahu. The State Department’s description of the meeting afterward suggested that the cease-fire agreement would be a broad one, dealing not just with the halting the violence, but with finding “a sustainable outcome that protects Israel's security and improves the lives of civilians in Gaza.”

A senior Israeli military official told McClatchy that the military had accomplished “all it could” given the situation and that “deterrence had been established,” a reference to the Israeli claim that its assault had been intended to stop rocket attacks from Gaza.

Hamas officials in Gaza told McClatchy they “had fought brave and well” and that their rockets had “struck deep into the heart of Israel,” bringing war to a part of Israel that previously had been immune to Gaza-based violence.

One Hamas official who’s been privy to the cease-fire talks in Egypt but who wasn’t authorized to speak to members of the news media said, “Hamas has shown itself to be strong and powerful. We stood up to Israel and achieved success.” He added that a string of visits by foreign officials from across the Arab world had given Hamas “a boost to keep going.”

Clinton’s hastily arranged trip to the region, which will include a joint appearance with Egyptian President Mohammed Morsi and U.N. Secretary General Ban Ki Moon in Cairo Wednesday afternoon, may have given both sides new incentive to reach a deal without appearing to lean on the United States. After five days, the talks had appeared to have stalled.