Prime Minister Lars Lokke Rasmussen said he was concerned that “democratic principles are under great pressure” in Turkey.
The rallies aim to encourage a large number of Turks living in Europe to vote yes in a referendum expanding the president’s powers.
Ties between the Turkish and Dutch leaders became particularly strained at the weekend after two Turkish ministers were barred from addressing rallies in Rotterdam, with one of them escorted to the German border.
Mr Erdogan likened the Netherlands to “a banana republic”, demanded international organisations impose sanctions on the Netherlands, and accused countries in the West of “Islamophobia”.
“I have said that I had thought that Nazism was over, but I was wrong. Nazism is alive in the West,” he added.
On Sunday, Mr Rutte demanded Mr Erdogan apologise for likening the Dutch to “Nazi fascists”.
“This country was bombed during the Second World War by Nazis. It’s totally unacceptable to talk in this way.”
The Netherlands would have to consider its response if Turkey continued on its current path, he added.
Meanwhile, German ministers also appeared to harden their rhetoric against Turkey.
Despite Chancellor Angela Merkel saying her government was not opposed to Turkish ministers attending rallies in Germany, as long as they are “duly announced”, her interior minister said he was opposed to Turkish political gatherings in Germany.
Separately, Finance Minister Wolfgang Schaeuble said Turkey had “destroyed the basis for further progress in co-operation”.
Sweden’s foreign ministry said it was not involved in the decision and that the event could take place elsewhere.
What is the row about?
Turkey is holding a referendum on 16 April on whether to turn from a parliamentary to a presidential republic, more akin to the United States.
So a number of rallies have been planned for countries with large numbers of eligible voters, including Germany, Austria and the Netherlands.
Why are countries trying to prevent the rallies?
Austrian Foreign Minister Sebastian Kurz said Mr Erdogan was not welcome to hold rallies as this could increase friction and hinder integration.
Germany in particular has been critical of the mass arrests and purges that followed – with nearly 100,000 civil servants removed from their posts.