Hamas has spoken of an easing of Israel's blockade on the 40-km (25-mile) slice of Mediterranean coast that is home to 1.7 million people. It may count on some sympathy from Mursi, though Egypt's first freely elected leader, whose Muslim Brotherhood inspired Hamas's founders, has been careful to stick by the 1979 peace deal with Israel struck by Cairo's former military rulers.
Clinton, who broke off from an Asian tour with President Barack Obama and assured Netanyahu of "rock-solid" U.S. support for Israel's security, spoke of seeking a "durable outcome" and of Egypt's "responsibility" for promoting peace.
She repeated international calls for the kind of lasting, negotiated, comprehensive Israeli-Palestinian settlement that has eluded the two peoples for decades - something neither of the two warring parties seems seriously to be anticipating.
"In the days ahead, the United States will work with our partners here in Israel and across the region toward an outcome that bolsters security for the people of Israel, improves conditions for the people of Gaza and moves toward a comprehensive peace for all people of the region," Clinton said.
"It is essential to de-escalate the situation in Gaza. The rocket attacks from terrorist organisations inside Gaza on Israeli cities and towns must end and a broader calm restored.
"The goal must be a durable outcome that promotes regional stability and advances the security and legitimate aspirations of Israelis and Palestinians alike."
Netanyahu, who has appeared in no immediate rush to repeat the invasion of winter 2008-09 in which over 1,400 Palestinians died, said: "If there is a possibility of achieving a long-term solution to this problem with diplomatic means, we prefer that.