Four boys playing football have been killed in Gaza by Israeli air strikes, according to Palestinian officials, as Israel responded to the death of a man from a barrage of rocket attacks with a bloody escalation of violence.

At least 16 Palestinians – including the four children – were killed yesterday as Israel responded to the deadly attacks the previous day.

While the Israeli military said it had been targeting militants and rocket-launching squads, the officials said the boys were playing football close to their homes in Jabalya, northern Gaza.

A relative, Ahmed Dardouna, 42, said the family had located the bodies of the boys – reportedly two brothers and their cousins – at a local hospital after they failed to return home. Rocket fire from militants into Israel continued during the day lightly injuring two Israelis and forcing the Israeli Interior Minister, Avi Dichter, to take cover during a visit to the border town of Sderot. His bodyguard was one of those injured, in a rocket attack launched before the minister arrived. Four rockets reached Ashkelon, 12 miles north of Gaza.

Palestinian officials said that while the majority killed were militants, 10 civilians were among the total of at least 27 killed over the past two days – seven of which were children.

They said that one Palestinian was killed and four wounded in a helicopter attack on a police roadblock 150 yards from the home of the de facto Hamas prime minister Ismail Haniyeh.

Although Mr Haniyeh is said to have gone into hiding and was not at home at the time, the attacks were locally interpreted as a warning to Hamas's political leadership because the coastal area of the Beach refugee camp where he lives is not used for launching rockets.

The latest outbreak of violence began on Wednesday when an airstrike killed five militants, described by Hamas as "five of our best fighters", as they drove in a van in the southern Gaza town of Khan Yunis.

Hamas claimed responsibility for a subsequent and unusually heavy retaliatory barrage of more than 40 Qassam rockets, one of which killed a 47-year-old father of four at the Sapir College in Sderot.

Israel swiftly responded with a series of air strikes beginning on Wednesday, killing two other children aged 10 and 11, according to Palestinian officials. A missile attack on the Ministry of Interior building on Wednesday evening killed a baby, Mohammed al-Burai, living nearby and caused severe destruction at the adjacent offices of the Oxfam-funded Palestinian Medical Relief Society.

Oxfam said the missile destroyed £15,000 of drugs in the pharmacy, a disabled centre catering for 400 chronically ill patients, an ambulance and a generator needed during the frequent power cuts in Gaza. Oxfam said last night: "We call on all parties to end military action which affects civilians and targets civilian infrastructure."

The Israeli Defence Minister, Ehud Barak, yesterday hinted at the possibility of a large-scale ground operation in response to the Qassam fire, saying: "We must prepare for continued escalation... We are not eager [to carry out such an operation], and not shying away from it. Israel will reach the perpetrators and Hamas will pay a price for its actions."

But the Israeli Prime Minister, Ehud Olmert, who met the US Secretary of State Condoleezza Rice in Tokyo, where he is on an official visit, said there were no "magic formulas" to eliminate rocket fire. He added: "We are suffering painful blows, but are returning more painful blows. We will continue fighting in order for the danger to the residents to end."

One of the Palestinians killed was Hamza al-Haya, the son of a senior Hamas Palestinian legislative council member, Khalil al-Haya, and said by the Islamic faction to have commanded a rocket-launching squad in northern Gaza. Visiting the morgue at Gaza, Mr Haya said: "I thank God for this gift. This is the 10th member of my family to receive the honour of martyrdom."

There were unconfirmed reports that a fifth child was also killed in Jabalya last night. As the Palestinian Foreign minister, Riyad al-Malki, condemned both the Qassam attacks and the Israeli military actions, Nabil Abu Rdeineh, spokesman for the Palestinian President, Mahmoud Abbas, said yesterday's air strikes "meant only one thing: the Israeli government... aims to destroy the peace process".

Ms Rice said: "I am concerned about the humanitarian condition there and innocent people in Gaza being hurt. We have to remember that the Hamas activities are responsible for what has happened in Gaza."

Palestinians claimed the air strikes included the use of F16s and Apache helicopters.