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Israeli embassy staff in Jordan have returned to Israel after a deadly incident at the complex on Sunday.

A statement from Israel's government said among those to return is a security officer, who had shot dead a Jordanian who had stabbed him.

Israel said all its diplomats were safe.

The incident led to a brief standoff as Jordanian authorities sought to question the guard, who Israel said had diplomatic immunity.

Prime Minister Benjamin Netanyahu has spoken to the ambassador and the guard, congratulating them on their handling of the situation.

The Jordanian had attacked him with a screwdriver at a residence used by the embassy in Amman, according to the Israeli Ministry of Foreign Affairs.

A second Jordanian was inadvertently killed in the gunfire, it said.

Israel thanked Jordan for its "close cooperation" over the last 24 hours.

The incident was one of the most serious between the two countries since they signed a peace treaty in 1994.

Jordanian police had said they wanted to question the guard, but Israel said he had immunity from investigation or arrest under the Vienna Convention of 1961.

The suspected attacker was named in local media as Mohammed Zakaria al-Jawawdeh, a 17-year-old carpenter.

The second Jordanian, who died from his wounds in hospital, was identified as the building's landlord.

The incident came at a time of heightened tension in the region over a Jerusalem holy site.

On Friday, thousands of Jordanians protested in Amman against Israel over the installation of metal detectors outside a site sacred to both Muslims and Jews in East Jerusalem.

Jordan, which occupied East Jerusalem from 1949 to 1967, is the custodian of the site, which is known to Muslims as Haram al-Sharif and to Jews as the Temple Mount.

Tensions between Israelis and Palestinians over the site have surged in recent days in response to the metal detectors, which were put in place following the killing nearby of two Israeli policemen.

Security cameras have now also been installed at a gateway leading to the site.