“The former Commander of the Air Force, General Ezer Weitzman, regarded as a hawk, stated that there was ‘no threat of destruction’ but that the attack on Egypt, Jordan and Syria was nevertheless justified so that Israel could ‘exist according the scale, spirit, and quality she now embodies.’...Menahem Begin had the following remarks to make: ‘In June 1967, we again had a choice. The Egyptian Army concentrations in the Sinai approaches do not prove that Nasser was really about to attack us. We must be honest with ourselves. We decided to attack him.’" (1)
“I do not think Nasser wanted war. The two divisions he sent to The Sinai would not have been sufficient to launch an offensive war. He knew it and we knew it.” (2)
Israeli cabinet minister Mordechai Bentov described the idea that Israel was facing extermination at the hands of incoming Arab armies as being, "invented of whole cloth and exaggerated after the fact to justify the annexation of new Arab territories." (3)
Israel also received intelligence reports from the US, saying that Egyptian military formations were defensive, and in anticipation of an Israeli attack. Israeli intelligence agreed with the Americans. (4)
It seems pretty clear that Arab mobilization aside, Israel never believed that they were going to be attacked, and thus, the claim of a legal preemptive-strike, at least in this sense, falls short.
Israel later claimed that their attack was justified in the face of Egypt's blockade of the Straits of Tiran, though this claim also has a few issues. There's little evidence that Egypt's blockade was regularly/stringently enforced (if at all). Debate about whether Egypt's closure of the straits was actually legal is also salient, as well as the fact that International Law only allows for a war in self-defense, in the face of an armed attack (something the Arabs never did).
There's also the fact that even if justified, Israel's attack must be proportionally equal to the level of threat they faced, which it wasn't.
I tend to view the war of 1967 as an aggressive attack by Israel, initiated primarily to deal a death knell to Nassir/Egypt (a rising threat to Israel's regional hegemony), and secondarily, to secure more land for "greater Israel".
Israel's frequent "changing of goalposts", also inspired little confidence in their claim as the defensive party. First they claimed they were attacked, but later had to admit they attacked first, but preemptively. When that position became untenable, they switched to the blockade as a cause for war, and so on.
Much of the world seemed to agree with the Arab position at the UN, though America and some European states didn't, which eventually led to the designation of the war as being caused by, "mutual aggression", which seems fair.
(1): The Fateful Triangle, Noam Chomsky. (2): Yitzhak Rabin, Israel’s Chief of Staff in 1967, in Le Monde. (3): Palestine & Israel: A Challenge to Justice, Quigley. (4): The Myth of Annihilation, Ryan.