In April Turkish voters will be asked whether they back a new constitution, said to transform the country from a parliamentary republic into a presidential one.
Among the proposed changes are those giving President Recep Tayyip Erdogan - who now holds a largely ceremonial role - sweeping new powers, including over the national budget.
The president would also have a greater say in the appointment of ministers and judges, as well as the authority to dismiss the parliament.
Around 1.4 million Turkish citizens live in Germany.
The Turkish government had planned to send ministers to address rallies in several cities in a bid to win support for its position among those eligible to vote in the referendum.
But German authorities last week withdrew permission for two events, citing security concerns.
President Erdogan criticised the decision at a public rally in Istanbul.
"Germany, you have no relation whatsoever to democracy and you should know that your current actions are no different to those of the Nazi period. When we say that, they get disturbed. Why are you disturbed? This is what you're doing."
Recep Tayyip Erdogan suggested The Netherlands has followed in Germany's footsteps.
"I thought the era of Nazism was over in Germany, but I see now that it's going on. Everything is clear. One of my ministers who was scheduled to meet your minister, wanted to give a speech. Why are you bothered? Now I see the Netherlands has made a similiar statement. You poor things."
Germany's Justice Minister Heiko Maas described the comments as "absurd, disgraceful and outlandish".
The incident has further strained relations between the two NATO member states, amid mounting public outrage over the arrest in Turkey of a Turkish-German journalist.
And there have been growing calls for a tougher response to Mr Erdogan's actions from Chancellor Angela Merkel.
Public polls accuse the German government of being too accomodating to Turkey, with which it signed an agreement to help stem the arrival of asylum seekers from Syria and elsewhere to Europe.
In Austria, meanwhile, Chancellor Christian Kern is calling for a European Union-wide ban on political campaigning by Turkish politicians.
And the Dutch far-right leader Geert Wilders says the entire Turkish cabinet should be banned from visiting the Netherlands.
"I think that coming here to advocate the change of the Turkish constitution that will only strengthen the Islamo-fascist leader Erdogan of Turkey more than parliament, Turkish parliament is the worst thing that could happen to us. So I would say if I would be prime minister today I would declare, at least until half of April when they have this referendum in Turkey, I would call the whole cabinet of Turkey persona non grata for a month or two."
