Israeli Prime Minister, Benjamin Netanyahu, ordered that air strikes on Gaza be as "surgical" as possible in a bid to stop the three days of renewed violence across the Gaza border getting out of control, officials said.

The apparent attempt to lower the temperature of the conflict triggered by last Thursday's attack by gunmen in Rafah which left eight Israelis dead came during a day of behind-the-scenes diplomatic activity – involving the Egyptian government among others – aimed at securing an agreed truce or "lull" in the cycle of Israeli air strikes and Palestinian rocket attacks.

Last week's attack came a few days after Egypt's ruling military council – which took power after ex-president Hosni Mubarak was toppled in February – sent hundreds of tanks to the North Sinai following outbreaks of violence in its provincial capital, Al-Arish, about an hour's drive west from Rafah.

Israel which is also striving to restore relations with Egypt, appeared for now to be resisting increasingly bellicose calls from opposition politicians for a much wider military operation in Gaza.

Shaul Mofaz, a prominent member of the Kadima party who chairs the influential Knesset Foreign Affairs and Defence Committee, said Israel needed to take further steps against Gaza's ruling faction Hamas and needed to "topple their infrastructures".

According to his spokesman, Mark Regev, Mr Netanyahu told officials that Israeli air strikes had two aims: to stop attacks on southern Israeli families within reach of rockets from Gaza, which killed a man in Beersheba on Saturday night; and to target those responsible for the rocket attacks.

At least 15 Palestinians – most of them militants but also including a doctor and three children under the age of 14 – have been killed in Israeli air strikes since Thursday's attack.