Both Saudi Arabia and Israel view arch-foe Iran as a main threat to the Middle East, and increased tension between Tehran and Riyadh has fuelled speculation that shared interests may push Saudi Arabia and Israel to work together.
It is the latest dramatic twist in weeks of turmoil in the region, which followed an unexpected purge of Saudi princes and officials by crown prince Mohammad bin Salman, who has also increasingly locked Saudi Arabia on a path to confrontation with Iran.
He replied: "We have ties that are indeed partly covert with many Muslim and Arab countries, and usually (we are) the party that is not ashamed".
Israel's Defense Minister Avigdor Liberman on Saturday called on the Arab world to normalize ties with the Jewish state and to present a united front against Iran's belligerence in the region.
Asked if Israel has recently shared intelligence with Saudi Arabia, Eizenkot said, "we are prepared to share information if there is a need to do so".
"There's an opportunity to form a new global coalition in the region with President Trump", Eisenkot added.
While Israel and Saudi Arabia do not have diplomatic relations, they have been edging closer recently, including a visit past year by a retired Saudi general heading a delegation seeking to encourage better relations.
Lt. Gen. Gadi Eizenkot told the privately Saudi-owned Elaph in what it said was his first interview with an Arabic newspaper that Israel had no plans to attack Lebanon's Hezbollah group. "We will not allow any Iranian presence".
Earlier this month, Lebanese Prime Minister Saad al-Hariri announced his resignation in a surprise move that had plunged Lebanon into a new political crisis.
"Iran seeks to take control of the Middle East, creating a Shiite crescent from Lebanon to Iran and then from the [Persian] Gulf to the Red Sea", he said. We respect the will of the other side: when connections develop, whether it is with Saudi Arabia or other countries...
At the same time, the chief of the General Staff of the Israeli Armed Forces noted the readiness of Tel Aviv to exchange intelligence with Riyadh to neutralize threats from Tehran.