I don't think it's fair to say that removing the Jews from Europe is a goal of Zionism - there's a big difference between deportation and emigration. Even the mass-migration of Jews from Europe is not really a goal of Zionism, but rather a side-effect.
Well, no thats not true, or at best, its semantics. The concepts of European emigration and a Jewish national homeland are utterly entwined - one could not exist without the other.
The fundamental premise of Zionism since the mid 1800's was that the only solution for a viable, communal Jewish existence lay in Eretz Israel, and the
negation of the Diaspora is a fundamental principle in nearly every current of zionism. The prime goal of the Jewish National movement from its foundation in Russia and Eastern Europe was to promote immigration to Palestine. The grand project of Herzel, the world Zionism Congress and Political zionism was to promote immigration and to find a 'corner' (be it in Europe, Uganda or elsewhere) for all Jews to emigrate to and establish a state. Herzel even suggested that anti-semitism be promoted and used as tool to aid this process:
"It would be an excellent idea to call in respectable, accredited anti-Semites as liquidators of property. To the people they would vouch for the fact that we do not wish to bring about the impoverishment of the countries that we leave. At first they must not be given large fees for this; otherwise we shall spoil our instruments and make them despicable as 'stooges of the Jews.' Later their fees will increase, and in the end we shall have only Gentile officials in the countries from which we have emigrated. The anti-Semites will become our most dependable friends, the anti-Semitic countries our allies."
And of course you have cultural zionists like Brenner and Kaufmann with borderline anti-Semitic descriptions of diaspora Jews as 'slovenly', 'depressed', 'disfigured' - positives for Brenner as he saw despair as a primary motivation for Zionism & immigration - in fact the similarities between Zionist portrayal of pacifist, parasitic, diaspora Jews and the basic tenets of European anti semitism have been well documented.
Jabotinsky, the paragon of revisionist Zionism provides probably the most eloquent summary:
"Eliminate the Diaspora, or the Diaspora will surely eliminate you."
This all continued in the post '48 period, with Israeli efforts to promote immigration from the arab nations including grenade attacks at Iraqi synagogues, military & political wrangling with other Arab (& even African) states, repeated public campaigns in the US, subsidised Aliyahs - amongst countless other examples. Anti diaspora ideas have never gone away, and have been periodically revived - primarily to counteract the 'demographic threat' of Palestinian Arab citizens of Israel. 'Diaspora Jew', coined as an insult in opposition to the dynamic, muscular 'Mensch' of Zionism at the start of the last century is still, today used as a synonym for 'self-hating Jew' in Israel today.
You could write a book on the subject - in fact, several dozen have been written, but I think that gives some kind of overview.
As for the Nazis - apart from The Haavara Agreement with German Zionists, they also famously planned (and nearly executed) the Madagascar plan - the forcible transfer of European Jews to Madagascar, and then again with Generalplan Ost - the transfer of jews to the Soviet Union via Poland.
So - the transfer of European Jewry, forcibly or voluntarily was a shared goal of Zionism and Nazism. The Nazis wanted to remove Jews from Europe and Zionists wanted European Jews to leave Europe for Israel. The motivations are poles apart, but the goal was more or less identical.