Barack Obama is pressing the Egyptian leadership to help de-escalate the bloody conflict in Gaza amid concern that a further ratcheting up of violence, such as a major Israeli ground assault, could damage the peace accords between Cairo and Jerusalem.

The Egyptian prime minister, Hisham Kandil, is expected to lead a delegation to Gaza on Friday for talks with Hamas leaders on quelling the fighting which has escalated with the Israeli assassination of the Hamas military leader, Ahmed Jabari, and the firing of hundreds of rockets from Gaza into Israel. The fighting has now claimed civilian lives on both sides.

On Thursday, the conflict threatened to escalate further as the Israelis stepped up their intense bombardment of Gaza and Hamas continued its barrage of rocket fire into Israel.

Three people were killed and three children injured when a rocket hit their apartment block in the southern Israeli town of Kiryat Malachi. Another rocket struck close to the city of Rishon Lezion, 22 miles north of Gaza, one of almost 150 fired from the enclave on Thursday.

In Gaza, there was mounting concern about rising civilian casualties, with the ministry of health reporting that four children and a woman pregnant with twins were among the dead. It said 130 people had been wounded. By late afternoon, Israeli jets were continuing, and Hamas said the death toll had reached at least 19.

Israeli troops were reported to be moving south in preparation for a possible ground invasion of Gaza. However, the IDF spokeswoman, Avital Leibovitch, indicated that such a move was not imminent, saying it was only an "option" and that ground forces were "on alert".

One trigger for an order to launch a ground offensive could be a rocket landing in Tel Aviv, Israel's second-largest city. Gaza's militants have never reached the city, which is roughly 50 miles north of the strip. But on Thursday, a rocket fired from Gaza reportedly landed in the sea close to the city.

The Egyptian prime minister's visit to Gaza follows a call by Obama to the Egyptian president, Mohamed Morsi, on Wednesday in which the US president urged him not to take any action that could cause a further deterioration in relations with Israel and threaten the Camp David peace accords, according to diplomatic sources.

"The American focus is getting the Egyptians to pull Hamas back, and making sure the Egyptians themselves do not do anything precipitous that could seriously damage the peace accords," said a western diplomat in the region. "The Americans recognise that there has to be a certain latitude for Morsi because he faces his own pressures to take a tougher stand with Israel.

"But they want to make sure it doesn't get to a point where the peace accords are under threat, and an escalation in Gaza could push it down that path. Washington is co-opting the Egyptians into making peace not only to get Hamas to rein it in, but to stop the Egyptians themselves from taking it over the cliff."

The US secretary of state, Hillary Clinton, spoke to the Egyptian foreign minister, Mohamed Kamel Amr, to press home the message. Amr called on the US to put pressure on the Israelis to halt the assault on Gaza, the foreign ministry said.

Morsi is maintaining a firm line in public, pulling the Egyptian ambassador out of Israel and calling an emergency meeting of the Arab League.

"Israel needs to understand that we won't accept the aggression that negatively affects the security and stability of the region," Morsi said on Thursday at the end of a meeting on the crisis.

The Egyptian foreign ministry handed a letter to the Israeli embassy in Cairo expressing "strong condemnation" of the military strikes on Gaza.

Obama also spoke to the Israeli prime minister, Binyamin Netanyahu, on Wednesday.

Aaron David Miller, a former US Middle East negotiator and now vice-president of the Wilson Centre, said he expects Obama to have appealed to the Israelis not to push the assault too far.

"The message to the Israelis is clearly restraint, give the Egyptians time to work this. Nobody has an interest in escalation," he said.

Publicly, the US administration has firmly backed Israel over the Gaza conflict.

The White House blamed Hamas and other armed Islamic groups for "cowardly" attacks and said in a statement about the president's call to Netanyahu that Obama reiterated "the United States's support for Israel's right to self-defence in light of the barrage of rocket attacks being launched from Gaza against Israeli civilians".

Miller said Obama will remain unwavering in his support of Israel – not just in Gaza but on diplomatic issues such as opposition to the Palestinian president, Mahmoud Abbas's plan to ask the UN general assembly to recognise Palestinian statehood later this month – despite criticism from Morsi that Washington siding with the Jewish state has damaged US interests in the Middle East.

"If Obama has any hope of promoting an Israeli Palestinian initiative down the road, he's going to have to remain in lock step with the current Israeli government on two issues – opposing Abbas's efforts on recognition New York and taking a very very tough line on Israeli security," said Miller. "There'll be latitude in giving the Israelis a lot of leeway in terms of the disproportionality of whatever response they undertake in Gaza."

But behind the scenes Obama is having to tread carefully with Morsi who has been more forcefully critical of Israel than his predecessor, the deposed ruler, Hosni Mubarak.

Morsi has said Egypt will continue to respect the peace accords but has also said Israel has not fulfilled its obligations – a claim that has caused concern in Washington that the long term stability of the agreement may be undermined.

In criticising the US for taking sides with Israel against the Palestinians, the Egyptian leader singled out Washington's "failure" to ensure Israel's compliance with what he said is a requirement in the Camp David agreement for it withdraw from the occupied Palestinian territories.

"As long as peace and justice are not fulfilled for the Palestinians, then the treaty remains unfulfilled," he told the New York Times in September.

The Egyptian president has also been strongly critical of Obama's failure to live up to the president's early promise of a shift in US policy in the Middle East and warned that it continues to foster anti-American sentiment in the Arab world.

Obama has contributed to the cooling of the relationship after the close ties between the Mubarak regime and the US by saying that he regards the present Egyptian administration as neither an ally nor an enemy.

But Miller said that Obama is also in a position to exert pressure on Morsi to ensure the peace treaty is not undermined by the Gaza conflict or broader political pressures, not least over Egypt's position as one of the largest recipients of US aid.

"We've got a $4.5bn IMF loan to Egypt under consideration and Morsi wants our help with that. Why is there massive assistance to Egypt? It grew out of the peace treaty. Without that peace treaty, the chances of our treating Egypt as a special case are going to diminish dramatically," he said.

"Morsi has a much more proactive agenda which is to actively work to make an already cold peace colder. Ultimately he could get away with it, to redefine the nature of the Egyptian-Israeli relationship which he doesn't like. The potential upside is that if Morsi's role is a positive one in defusing this crisis, it would serve to inject a fair amount of stability in the Egyptian-Israeli relationship right now because under a Muslim Brotherhood president, every day that peace treaty exists it's legitimised."