The interview - described by the Israeli newspaper Haaretz as "unprecedented" - would have required Israeli political approval at the highest level, given the lack of diplomatic relations with Saudi Arabia.
Eisenkot added that "Under U.S. President Donald Trump there is an opportunity to form a new worldwide alliance in the region".
Bahrain's foreign minister said on Sunday that Lebanon is under the "total control" of Hizballah, in a speech at an Arab League meeting called by Saudi Arabia.
Arab foreign ministers on Sunday condemned Iran's intervention in the Arab countries' affairs, according to a statement released on Sunday. We have ties, which are partially secret, with many Muslim and Arab states... He described Sadat as "a courageous leader, who stood against the tide and paved the way for other Arab leaders to recognize the importance of strategic ties with Israel".
Last week, the Israeli military chief, Lieutenant General Gadi Eizenkot, told an Arabic language online newspaper that Israel was ready to share "intelligence information" with Saudi Arabia, saying their countries had a common interest in standing up to Iran.
Sunni Muslim powerhouse Saudi Arabia and Iran, the predominant Shiite power, have for decades stood on opposing sides of conflicts in the Middle East including in Syria and Yemen.
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Formal ties do not seem likely due to Israel's continuing occupation of Palestinian territory and the lack of progress in peace efforts, but behind-the-scenes cooperation has opened up in various areas, a number of experts and officials have said.
While Israel and Saudi Arabia do not have diplomatic relations, they have been edging closer recently, including a visit a year ago by a retired Saudi general heading a delegation seeking to encourage better relations.
Ibish said that given the mutual threat perceptions shared by Israel and Gulf Arab countries, it is unlikely that covert ties aren't developing.
The minister, a long-time ambassador to the U.S. appointed as foreign minister two years ago, said that 80 Iran-manufactured missiles had been launched from Yemen since the conflict broke out in the country in March 2015.
Saudi Arabia has since 2015 been leading an Arab coalition fighting on the side of Yemen's internationally recognized government.
Saudi Arabia's powerful Crown Prince Mohammed bin Salman has accused Tehran of "direct military aggression" against the kingdom by supplying the Yemen rebels with ballistic missiles, but Tehran has denied any involvement.
In 2016, former Saudi general Anwar Eshki visited Israel, where he met Israeli legislators, to promote - as he has at various academic forums - the Saudi peace initiative.