The two leaders disagreed on several important issues
German Chancellor Angela Merkel has repeated her belief that Turkey is not suitable for full EU membership, on the first day of an official Turkish visit.
Turkey should not view her offer of a "privileged partnership" - rather than full membership - negatively, she said.
Turkey, which began negotiations to become a member in 2005, has dismissed that offer as an insult.
There were also disagreements over Iran, Cyprus and educating Turkish children in Germany in Turkish.
Germany is Turkey's biggest trading partner, its biggest foreign investor, its biggest source of tourist revenue, and nearly three million Turks live in Germany.
But, says the BBC's Jonathan Head in Istanbul, for two countries with such deep-rooted historical and economic ties, Germany and Turkey struggled to find a common voice during Mrs Merkel's visit.
'Shifting the goalposts'
On every important issue, Chancellor Merkel and Turkish Prime Minister Recep Tayyip Erdogan seemed to be at odds with each other, our correspondent adds.
Angela Merkel wants Turkey to have a privileged partnership with the EU
After months of avoiding the subject, Chancellor Merkel chose this moment to revive her idea of offering Turkey what she calls a privileged partnership with the EU, rather than full membership.
Mrs Merkel has stressed that she does see integration as possible in up to 28 of the 35 so-called chapters of EU law with which Turkey has to comply before it can become a full member of the union.
Speaking at a joint press conference in Ankara, she said: "The rules of the game have changed" since 2005.
"The (accession) negotiations are an open-ended process. We should now pursue this open-ended process."
Mr Erdogan said nothing - but has in the past expressed his outrage over what he calls "shifting the goalposts".
The German proposal has been firmly rejected by the Turkish government as a breach of the terms agreed when membership negotiations began five years ago.
"Such a thing as privileged partnership does not exist," said Egemen Bagis, Turkey's minister for European affairs.
"So we do not take that option seriously because there is no legal foundation of it. At times I feel insulted for being offered something which does not exist."
Turkey's sometimes fraught relationship with the European Union will not be helped by this visit.
On Iran, Mrs Merkel wants Turkish support for international sanctions, which Mr Erdogan argues are counter-productive and hypocritical.
Turkey has recently strengthened its relations with Iran and opposes the tougher sanctions threatened by Western governments.
"We are of the view that sanctions is not a healthy path and... that the best route is diplomacy," said Mr Erdogan at the joint news conference.
The two leaders also disagreed about Cyprus, with the German chancellor calling for the issue of Turkey's refusal to recognize the government on the divided island to be resolved quickly.