Asked about Dundar, who was convicted by Turkey in 2016 of revealing state secrets after his newspaper published photographs suggesting that Turkey's intelligence agency was involved in sending weapons to Syrian rebels, Erdogan called him a "convicted spy" who should be extradited.
The two North Atlantic Treaty Organisation allies agreed to jointly discuss the Syria crisis in a meeting next month with Russian President Vladimir Putin and Emmanuel Macron of France, Merkel said at a joint press conference. Turkey accuses Gulen of orchestrating the coup attempt, which he denies.
Erdogan also used Friday's news conference to call for the German authorities to extradite all members of the Kurdish PKK and the Gulen movement - both considered terrorist organizations by Turkey - now in Germany.
At the same conference, Mr Erdogan said Turkey aims to win easier access to the European Union for its citizens and his government would move to meet the EU's criteria for achieving visa liberalisation.
"Mr Erdogan lands in Berlin, journalists land in prison", read a poster from Reporters without Borders, a taster of more expected to come.
Merkel said she and Erdogan talked about bilateral economic co-operation and stressed that "Germany has an interest in an economically stable Turkey".
And as Erdogan was whisked through a locked-down city to his hotel by the Brandenburg Gate, allegations that a Berlin police officer had been spying on exiled dissidents for Turkish security services highlighted the two sides' mutual mistrust.
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German Chancellor Angela Merkel is pressing Turkish President Recep Tayyip Erdogan for the release of Germans still detained in Turkey in the wake of a crackdown following the country's 2016 coup attempt.
Merkel said she didn't want to discuss specifics, but that it was "no secret" that Turkey and Germany differed in the case of Dundar, who fled to Germany. The most prominent have been released, but five remain in Turkish custody.
But she added: "There are still profound differences between us over press freedom and the rule of law".
Dr Merkel herself has repeatedly stressed the importance of good relations with Ankara, a partner she relies on to help stem the flow of migrants to Europe.
Rumors were circulating Friday morning that Erdogan was threatening to boycott the news conference if Can Dundar, former editor-in-chief of Cumhuriyet newspaper, a Turkish news outlet critical of Erdogan, were to attend as planned.
Erdogan is eager to use his three-day visit to rebuild trade ties.
Erdogan started his state visit with military honours at the Bellevue Palace where he held talks with President Frank-Walter Steinmeier.