"Here and now, I have impression that the membership process is not the right framework to have a successful dialogue with Turkey," Michel said in an interview with the Associated Press.
Austria is also seeking to end Turkey's membership bid, and a growing group of countries say they've realized acting as if Turkey is still a constructive partner would amount to a charade.
Erdogan recently won a referendum that expands his powers, and he has had equally harsh words for the EU. He has also said he may hold a referendum on whether the country should continue its membership efforts.
The suppression of an attempted coup last summer and the resulting purge of tens of thousands of people from government jobs have been widely criticized in Europe as a move toward authoritarianism. Erdogan's campaign for the referendum giving him more power further undermined whatever goodwill was left.
"The past months made a lot clear about the developments in Turkey and the way in which the referendum campaign was held. It was clearly an anti-European campaign. So, at a certain point for the European Union, masochism must have its limits," Michel said. "This is not simply becoming member of an economic or political club, it is also becoming member of a set of fundamental values."
Turkey and Europe are already politically intertwined because of a shared membership in NATO and strong commercial links. The EU also clinched a deal in which Turkey gets billions of euros in aid in return for keeping refugees - mostly Syrian, who would otherwise would be pouring into Europe in huge numbers.
"Turkey will always be our neighbor so there is a great strategic interest to have a dialogue," Michel said.
Early this week, Erdogan says his country will have "nothing to discuss" with the EU unless the membership drive gets going again.
Michel said that "we are approaching the moment of truth with Turkey. I think more and more that the EU membership process is ever more at a dead end.
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