Israeli Prime Minister Benjamin Netanyahu has warned of a "prolonged" military campaign in Gaza.
Mr Netanyahu said Israel would continue to act until it had achieved its aim of destroying militant group Hamas' network of underground tunnels.
It comes amid an upsurge of violence in Gaza and southern Israel, despite a UN plea for a cessation of hostilities.
Fighting continued in the early hours of Tuesday, with attacks reported from both sides.
Officials say over 1,070 Palestinians, most of them civilians, have been killed in the fighting since 8 July. Israel says 48 of its soldiers and three civilians - two Israelis and a Thai worker, have been killed.
Early on Tuesday, Israeli aircraft fired at the house of the former Hamas Prime Minister, Ismail Haniyeh, his son and Gaza officials said.
The house was thought to be empty at the time, Reuters news agency said.
Hamas said that its TV station, Al-Aqsa TV, was also targeted, reports said.
Meanwhile, the Israel Defense Forces (IDF) said that three rockets were fired from Gaza, with one rocket successfully intercepted and two hitting open spaces in central Israel.
On Monday, explosions in Gaza City reportedly killed 10 people, including children.
Israel said that five of its soldiers died on Monday, and five Hamas militants were also killed inside Israel.
Mr Netanyahu described Monday as a "painful day", adding: "We have to have patience."
"We will continue to act aggressively and responsibly until the mission is completed to protect our citizens, soldiers and children," he said.
On Monday, UN Secretary General Ban Ki-moon criticised both sides for firing into civilians areas, and called for an immediate, unconditional humanitarian ceasefire in Gaza during the Muslim Eid al-Fitr holiday.
"In the name of humanity, the violence must stop," he told reporters, adding that Gaza was in a "critical condition".
His spokesman later added that Mr Ban was concerned at reports that leaflets had been dropped by the IDF warning residents in the northern Gaza Strip to evacuate to Gaza City.
"If true, this would have a further devastating humanitarian impact on the beleaguered civilians of those areas of the Gaza strip, who have already undergone immense suffering in recent days," his spokesman said.
At least 10 people - eight of them children - were killed in blasts in Gaza City on Monday afternoon, Palestinian health officials said.
They said the 10 were killed by Israeli missile strikes, but Israel says the explosions were caused by rockets misfired by "terrorists".
Four Israeli soldiers were killed and another 10 injured when a mortar shell hit the Eshkol district in southern Israel on Monday.
The Israeli military said the five Hamas militants who died had entered Israel via a tunnel from Gaza and opened fire on Israeli troops, who returned fire.
Rocket fire and air strikes between the two sides increased after the abduction and killing of three Israeli teenagers in June, which Israel blamed on Hamas and which led to a crackdown on the group in the West Bank. Hamas denied being behind the killings.
Tensions rose further after the suspected revenge killing of a Palestinian teenager in Jerusalem on 2 July. Six Jewish suspects were arrested over the youth's abduction and murder.
On 8 July, Israel launched Operation Protective Edge, an offensive against Hamas in Gaza, after a surge in rocket fire.
On 18 July, it extended operations with a ground offensive, saying it was necessary to destroy tunnels dug by militants to infiltrate Israel.
Since then, there have been hundreds of air strikes and hundreds of rockets have been fired.
The Gaza Strip, sandwiched between Israel and Egypt, has been a recurring flashpoint in the Israel-Palestinian conflict for years.
Israel occupied Gaza in the 1967 Middle East war and only pulled its troops and settlers out in 2005. Israel considered this the end of the occupation, but it still exercises control over most of Gaza's borders, waters and airspace. Egypt controls Gaza's southern border.
Credit: The BBC