In 1973 [British MP Christopher Mayhew] offered £5,000 to anyone who could produce evidence that Nasser had stated that he sought to "drive the Jews into the sea". Mayhew repeated the offer later in the House of Commons (Hansard, 18 October 1973) and broadened it to include any genocidal statement by an Arab leader (The Guardian, 9 September 1974), while reserving for himself the right to be the arbiter of the authenticity of any purported statements as well as their meaning. Mayhew received several letters from claimants, each one producing one quotation or another from an Arab leader, all of which Mayhew deemed to be fabricated. One claimant, Warren Bergson, took Mayhew to court. The case came before the High Court in February 1976. Bergson was unable to offer evidence of Nasser's alleged statement and acknowledged that, after thorough research, he had been unable to find any statement by a responsible Arab leader that could be described as genocidal.The Internet is filled with this anecdote about the court case, and after much searching it appears to be true (see page 3 here. )
Hassan al-Banna, founder of the Muslim Brotherhood, was quoted by the New York Times in 1948 as saying "If the Jewish state becomes a fact, and this is realized by the Arab peoples, they will drive the Jews who live in their midst into the sea." That wasn't about Israelis but about Jews in Arab countries.
So did Arabs ever use that phrase concerning the Jews of Israel, or is it a myth?
It does not appear to be a myth.
This book by (unreliable) author Gilbert Achcar says that Nasser quoted the phrase in an ironic way:
The Gamal Abdel-Nasser Foundation's Web site, created in collaboration with the Bibliotheca Alexandrina,'° contains 1,359 declarations and transcriptions of speeches or interviews by the founder of modern Egypt. This electronic resource makes it considerably easier to analyze Nasser's discourse. There one finds an address delivered by Bikbachi (Colonel) Abdel-Nasser before Alexandria's Palestine Club on December 13, 1953.... In it, the future president offers an ironical comment on the Arab attitude during the period when the Zionist plan was being translated into action under the British mandate: "At our meetings and in our speeches, we said we were going to throw the Jews into the sea, and felt reassured after every speech. Then we all went back home:"
So while Nasser may not have made that call directly (and he actually denied ever saying that,) it seems to have been a common enough threat among Arabs themselves.
But did any Arab leader make that threat?
Yes. Here is a statement by Dr. Fadhil Jamali, Iraqi Representative to the United Nations, speaking to the Arab League, February 6, 1955:
I asked them (Arab League members) how Palestine was lost. It had been lost for two basic reasons: one, because we deluded ourselves by underestimating the power of our opponent and by thinking that the Jews were not powerful. The highest official in the League said that with 300 soldiers or North African Volunteers we could throw the Jews into the sea. The war started and His Excellency then said that with 3,000 North African Volunteers we could throw them into the sea. The second reason was that we thought that we were strong enough to face the world but the fact was that we did not estimate our own strength correctly. This then was the issue of Palestine. It seemed a trifling thing at the time but we did not know that behind the Jews of Palestine stood World Zionism with its resources in every major country.Jamali had no incentive to lie about this for his audience. So while this is second-hand evidence, it seems pretty strong.
UPDATE: The book O Jerusalem, by Collins and Lapierre, mentions that the phrase was used by a number of Arab leaders.
The first one to say he wanted to "throw the Jews int the sea," according to the book, was Kamal Irekat (almost certainly the same family as Saeb Erekat,) a police chief in Jerusalem.
It says that the Mufti of Jerusalem adopted the phrase.
Finally, the book attributes it also to Fawzi el Kaukji, an Arab League field commander.
(h/t David Sigeti)