The German federal prosecutor had announced earlier in the week the opening of an investigation into claims that Turkish agents have been spying on followers of Gulen, whom Turkish President Recep Tayyip Erdogan blames for a failed coup attempt last year.
Relations between NATO allies Germany and Turkey have been strained by disputes centred on human rights issues, especially since last July's failed coup and a subsequent crackdown on alleged conspirators.
The row deepened after Germany and the Netherlands blocked campaign events by Turkish ministers earlier this month ahead of a referendum vote to boost presidential powers. In an angry response, Erdogan accused both countries of
using "Nazi" methods.
More than 41,000 people in Turkey have been arrested over suspected links to Gulen's movement, and 100,000 fired or suspended from their jobs. Many of them are teachers, police, magistrates and journalists.
Boris Pistorius, interior minister of the German state of Lower Saxony, said on Tuesday that Erdogan's government had asked Berlin to help spy on about 300 alleged Gulen supporters in Germany.
On Friday the Sueddeutsche Zeitung daily and regional public television channels said that German prosecutors had last month begun investigating Halife Keskin, who heads the foreign relations department of the Turkish state religious affairs agency Diyanet.
According to the media reports, Germany suspects the agency of asking its members based overseas, notably imams in Germany, to gather information on people deemed close to Gulen.
The German Der Spiegel weekly reported Friday that Diyanet had asked Turkish consulates in 35 countries to gather information about Gulen supporters.
The magazine cited Australia, Nigeria, Mauritania and Mongolia as among the countries targeted.