CAIRO — Diplomats scrambled to avert a crisis in relations between Egypt and Israel on Saturday, and the Israeli government issued a rare statement of regret for the killing of three Egyptian security officers by an Israeli warplane.

Tensions between the two countries, which on Saturday led Egypt to announce that it would recall its ambassador from Tel Aviv, reached the worst point since the Camp David peace accords three decades ago, spurred by a burst of violence along their shared border in the Sinai Peninsula. A series of attacks there killed eight Israelis on Thursday; the Israeli government then retaliated against Gaza-based militants, and the three Egyptians died in the crossfire.

After Egypt’s announcement about its ambassador early Saturday, diplomats from other nations rushed to broker an end to the impasse between the Egyptians and the Israelis, a Western diplomat said, speaking on condition of anonymity because of the delicacy of the talks.

Breaking a customary silence on the Sabbath, the Israeli defense minister, Ehud Barak, released a statement saying, “We regret the deaths of members of the Egyptian security forces during the terror attack on the Israeli-Egyptian border.”