The Israeli army said an Egyptian assailant shot dead two soldiers "in the early morning... while they secured a military post at the Egyptian border" on Mount Harif, near the town of Mitzpe Ramon in the Negev desert.

The discovery of their bodies triggered a manhunt during which the third soldier was killed.

An army statement identified the assailant as an Egyptian policeman, saying he was killed by Israeli troops after being found "in Israeli territory".

The statement said an investigation was being carried out "in full cooperation with the Egyptian army".

A fourth Israeli soldier, a non-commissioned officer, was lightly wounded and evacuated to hospital, the military added.

An Egyptian army spokesman said "a member of the security forces... chased drug smugglers. During the chase, the security agent crossed the security fence (border)" and an exchange of fire took place.

Egypt's Defence Minister Mohamed Zaki later spoke with his Israeli counterpart Yoav Gallant to "discuss the circumstances of today's incident", the defence ministry in Cairo said.

Israel's defence ministry said Gallant "emphasized the importance of cooperating further in order to prevent such events in the future".

Egypt was the first Arab country to make peace with Israel following the Camp David accords of 1978, though there remains widespread popular opposition towards normalisation in Egypt.

Their shared border is largely peaceful, despite an insurgency in Egypt's Sinai Peninsula and occasional exchanges of fire between drug smugglers and Israeli forces.

Israel's army had initially declined to confirm media reports of the first two deaths, while the families were being notified.

The two soldiers, a man and a woman, were "killed by live fire adjacent to the border", it said in a statement.

The female soldier was identified as Lia Ben Nun, 19, and the male was 20-year-old Ori Izhak Iluz. The third soldier killed later was Ohad Dahan, 20.

- Border incidents -

A barrier runs along the Israel-Egypt border, and an army spokesman said Israel was looking into how it had been breached.

Hours before the border post attack, Israeli soldiers had foiled an attempted drug smuggling operation at the border, seizing contraband goods estimated to be worth 1.5 million shekels ($400,000), a spokesman said.

The spokesman added that no link had been established between the drugs seizure and the attack.

Israeli Prime Minister Benjamin Netanyahu described the incident as "severe and extraordinary" and said it would be "fully investigated".

His chief of staff, Herzl Halevi, said: "We are investigating the incident in a thorough, in-depth manner, together with the Egyptian Armed Forces."

There have been several previous incidents along Israel's border with Egypt.

In 2011, assailants who entered from the Sinai killed eight Israelis in a triple ambush north of Eilat. Pursuing Israeli forces killed seven attackers and five Egyptian police.

In 2012, an Israeli soldier and three militants who infiltrated from the Sinai were killed in a clash along the border.

In 2014, two Israeli soldiers on patrol were wounded by unidentified men who fired an anti-tank weapon from the Sinai during an attempt to smuggle drugs.

And in 2015, rockets fired from the Sinai hit southern Israel without causing any casualties. The Islamic State group later claimed responsibility.