As Israel’s bombardment of the Gaza Strip continues with no let-up in sight, lawmakers are focusing their attention on Egypt, warning that country’s new president to rein in the militant group Hamas or risk U.S. aid.

“Egypt, watch what you do and how you do it,” South Carolina Republican Sen. Lindsey Graham, a senior member of the Armed Services Committee, told NBC’s “Meet the Press” on Sunday. “You’re teetering with Congress on having your aid cut off if you keep inciting violence between Israelis and Palestinians.”

Congress appropriated $1.3 billion in military aid and $250 million in economic aid for Egypt in fiscal 2012, and the White House has asked for the same amount for fiscal 2013. Ever since it signed a peace treaty with Israel in 1979, Egypt has been the second largest recipient of foreign aid after the Jewish state.

Israel began its air and naval assault on the Hamas-ruled Gaza Strip on Nov. 14 after months of rocket fire against southern Israeli towns from the Palestinian territory. In response, Egyptian President Mohammed Morsi, a member of the Muslim Brotherhood — which counts Hamas as one of its branches — withdrew his ambassador from Israel in protest and sent his prime minister, Hisham Kandil, to visit the Gaza Strip in an unprecedented display of solidarity with Hamas.

Many lawmakers are looking for Egypt to play a mediator’s role in bringing about a cease-fire between the two sides.

“President Morsi insists that he rejects violence and is a force for moderation. Now is the time for him to prove it,” California Republican Ed Royce, likely the next chairman of the House Foreign Affairs Committee, said in a Nov. 15 statement.

An Israeli envoy was in Cairo on Monday to work with Egyptian officials on cease-fire proposals. Meanwhile, leaders of Egypt, Turkey, Qatar and Hamas also consulted over a cease-fire. Israel wants the rocket fire to stop, and Hamas wants Israel to lift its blockade of Gaza, a blockade that has been in place since Hamas took over the territory in 2007.

Ghaith al-Omari, executive director at the American Task Force on Palestine, said the fighting in Gaza reflects an attempt by Hamas to flex its muscles in the wake of the Arab Spring and an effort by Israel to disabuse Hamas of any illusions of greater power. He said Israel also does not want to give Morsi any time to establish foreign policies that are detrimental to Israel. The idea, al-Omari said, was to “force him to define his goals at an early stage.”

Lawmakers have been wary of Morsi ever since he won Egypt’s first democratic elections in June, replacing the ousted authoritarian Hosni Mubarak. More responsive to Egyptian public opinion, Morsi angered lawmakers in September when he withdrew police protecting the U.S. embassy from a rampaging mob protesting an American-made video mocking the Prophet Muhammad and waited three days before offering a tepid condemnation of the embassy attack.

As a result, Republicans Kay Granger of Texas, who chairs the House Appropriations subcommittee that oversees foreign aid, and Ileana Ros-Lehtinen of Florida, the outgoing chairwoman of the House Foreign Affairs Committee, slapped a hold on an administration plan for a $450 million cash transfer to Egypt to replenish the country’s dwindling cash reserves. They also oppose any additional money for a fund aimed at stimulating private investment in Egypt.

The hold remains in place despite administration appeals to lift it.

The last time Israel mounted a major military operation against the Gaza Strip was in late 2008 and early 2009. Using air, naval and ground forces, the Israelis killed some 1,400 Palestinians. Thirteen Israelis were killed.

The Israel-Gaza border remained relatively quiet until September, when militants in Gaza stepped up their rocket attacks to more than 600 per month. Armed with more sophisticated missiles from Iran, some rockets have reached as far as Tel Aviv and Jerusalem.

In the current round of fighting, the exchanges of fire have killed more than 100 Palestinians and three Israelis. Israel’s ground and naval bombardments have not silenced the rockets, and Israeli officials have moved tanks and troops to the Gaza border in preparation for a possible ground assault.

President Barack Obama took time out Monday from his trip to Asia to call Morsi and Israeli Prime Minister Benjamin Netanyahu.

“I think the president has been right on in supporting Israel’s right to defend itself,” said Sen. Joseph I. Lieberman, I-Conn.

Arizona Sen. John McCain, ranking Republican on the Armed Services Committee, suggested Sunday on CBS’ “Face the Nation” that Obama appoint former President Bill Clinton as a special envoy to revive moribund Middle East peace talks. In 2000, Clinton tried — unsuccessfully — to mediate a peace treaty between Israel and the Palestinians.

Emily Cadei and Niels Lesniewski contributed to this report.