Israel on Sunday buried three soldiers killed in a firefight at its usually secure border with Egypt and announced an "exhaustive and thorough" investigation into the deaths.

Egypt has blamed the fatal clash in the Negev desert on a member of its security forces who was also killed, saying he had crossed the border in pursuit of drug traffickers when he encountered the Israeli troops.

The Israeli army said an Egyptian "assailant" shot dead two soldiers at a post on the border. The discovery of their bodies triggered a manhunt during which the third soldier was killed as well as the suspected attacker.

Israel's border area with Egypt has not often been marred by violence since Egypt became the first Arab country to make peace with Israel following the Camp David accords of 1978.

As Israel mourned those killed the previous day, Prime Minister Benjamin Netanyahu said it had sent Egypt a "clear message", speaking at the opening of a cabinet meeting on Sunday.

"We expect that the joint investigation will be exhaustive and thorough," he said.

"This is part of the important security cooperation between us, which has benefitted both countries over the years."

The two Israeli soldiers whose bodies were found early Saturday at a guard post close to the Harif military base were identified as Lia Ben Nun, 19, and Ori Izhak Iluz, 20.

Ohad Dahan, also 20, was killed during the manhunt later on Saturday, the army said.

A fourth Israeli soldier, a non-commissioned officer, was lightly wounded, it added.

Ben Nun was laid to rest on Sunday in the central Israeli city of Rishon Lezion.

"I love you and miss you already," her sister Ofir said in her eulogy. "I don't know what I'll do now."

Iluz's funeral was held in the northern city of Safed, and Dahan was buried in southern town of Ofakim.

- 'Full investigation' -


Israeli media have raised questions over the shootings, particularly on how the assailant -- who has not been identified -- managed to cross the high barrier running along the border.

On Sunday, Egyptian military officers and defence ministry representatives arrived in Israel and began a joint investigation of the incident, a source close to the probe told AFP.

Speaking on condition of anonymity because of the sensitivity of the issue, the source said that while the Egyptian shooter was not believed to be affiliated with any Islamist organisation, he appears to have undergone religious radicalisation.

Netanyahu on Saturday promised a "full investigation" into the deaths, and senior government figures stressed the importance of cooperation with Egypt.

The Israeli army was conducting "a thorough investigation... in collaboration with the Egyptian Armed Forces", said the military chief of staff Herzi Halevi.

Defence Minister Yoav Gallant highlighted "the importance of the ties between the two countries" following a Saturday telephone call with his Egyptian counterpart, Mohamed Ahmed Zaki.

Zaki, meanwhile, underlined "the joint coordination to take the necessary measures to avoid the repetition of incidents of this kind in the future", according to a spokesman for Egypt's army.

The border between the two countries is generally calm, but has seen regular smuggling attempts.

In recent years, there have been exchanges of fire between smugglers and Israeli soldiers stationed 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.