Egypt's President Abdel Fattah al-Sisi has said officials were talking with both sides "around the clock" to ease the violence. A security source in Cairo said that Israel "has accepted" a ceasefire, adding that Cairo was waiting for the Palestinian response.
A source from Islamic Jihad said that "discussions are underway at the highest levels towards calm", but warned that "the resistance will not stop if the occupation's (Israel) aggression and crimes do not stop".
On Sunday, Islamic Jihad -- an Iran-backed group designated as a terrorist organisation by several Western nations -- extended their barrage to fire two rockets targeting Jerusalem, but they were shot down by the army.
The Israeli army has said the entire "senior leadership of the military wing of the Islamic Jihad in Gaza has been neutralised".
Muhammad Abu Salmiya, director general of the Shifa hospital in Gaza City, said medics were treating wounded people in a "very bad condition", warning of dire shortages of drugs and fuel to run power generators.
"Every minute we receive injured people," he said, earlier on Sunday.
In Gaza, run by the Islamist group Hamas, the health ministry said 31 people had been killed.
But Israel said it had "irrefutable" evidence that a stray rocket fired by Islamic Jihad was responsible for the deaths of several children in Gaza's northern Jabalia area on Saturday.
It was not immediately clear how many children were killed there, but an AFP photographer saw six dead bodies at the local hospital including three minors.
"We came running to the place and found body parts lying on the ground... they were torn-apart children," said Muhammad Abu Sadaa, describing the devastation in Jabalia.
The army said it had struck 139 Islamic Jihad positions, with the militants firing over 600 rockets and mortars, but with more than a hundred of those projectiles falling short inside Gaza.
Amid the high tensions, Jews in Israel-annexed east Jerusalem marked the Tisha Be'av fasting day Sunday at the Al Aqsa mosque compound, known in Judaism as the Temple Mount.
Some Palestinians shouted "God is greatest" in response, and an AFP photographer was briefly detained by Israeli police, but commemorations passed off without major incident.
Israel has said it was necessary to launch a "pre-emptive" operation Friday against Islamic Jihad, which it said was planning an imminent attack.
The army has killed senior leaders of Islamic Jihad in Gaza, including Taysir al-Jabari in Gaza City and Khaled Mansour in Rafah in the south.
Israel's Lapid called the killing of Mansour an "extraordinary achievement".
'Killing and wounding'
Daily life in the Gaza Strip has come to a standstill, with the sole power station shut down due to a lack of fuel after Israel closed its border crossings.
The United Nations humanitarian agency OCHA has warned of the "severe risk" to the "continuation of basic essential services".
"Each day we wake up to child and women martyrs... there is killing and wounding and people displaced," said Gaza City resident Abu Mohammed al-Madhoun, 56. "We hope that Israeli aggression will end".
In southern and central Israel, civilians were forced into air raid shelters. Two people were hospitalised with shrapnel wounds and 13 others lightly hurt while running for safety, the Magen David Adom emergency service said.
"It's tense, it's frightening," said Beverly Jamil, a resident of Ashkelon close to Gaza.
"Ashkelon's a ghost town -- it's a holiday, kids should be out playing."
Hamas's response to the violence remains critical, with spokesman Fawzi Barhoum offering the group's support to Islamic Jihad on Sunday, but stopping short of saying they would take part.
"The resistance in all its military wings and factions are united in this battle," Barhoum said.
Islamic Jihad is aligned with Hamas but often acts independently. Hamas has fought four wars with Israel since seizing control of Gaza in 2007, including the conflict last May.