Reports say the head of Turkey's intelligence service handed a list of people suspected of opposition sympathies to his German counterpart.
The list is said to include surveillance photos and personal data.
Germany and other EU states have banned local rallies in support of Turkish President Recep Tayyip Erdogan.
Turkish ministers have been seeking to campaign among ethnic Turks in a referendum on 16 April on increasing his powers.
Some 41,000 people have been arrested in Turkey since a coup was defeated in July of last year.
Many are suspected of following a movement loyal US-based cleric Fethullah Gulen. Turkey has accused him of organising the coup.
Turks in Germany have begun voting in a crucial referendum on boosting the powers of President Erdogan
According to Germany's Sueddeutsche Zeitung newspaper and several public broadcasters, the head of Turkey's intelligence service MIT, Hakan Fidan, handed Bruno Kahl a list of 300 individuals and 200 organisations thought to be linked to the Gulen movement at a security conference in Munich in February.
The apparent aim was to persuade Germany's authorities to help their Turkish counterparts but the result was that the individuals were warned not to travel to Turkey or visit Turkish diplomatic addresses within Germany, home to 1.4 million voters eligible to vote in the referendum.
Mr De Maiziere said the reports were unsurprising.
"We have repeatedly told Turkey that something like this is unacceptable," he said. "No matter what position someone may have on the Gulen movement, here German jurisdiction applies and citizens will not be spied on by foreign countries."
Recep Tayyip Erdogan wants Turks to back plans to scrap the role of prime minister and create an executive presidency
Police in the western state of North Rhine-Westphalia told German TV that the lists had to be taken seriously, and there was an angry response from senior German figures.
"Outside Turkey I don't think anyone believes that the Gulen movement was behind the attempted putsch," said German spy chief Hans-Georg Maassen.
"At any rate I don't know anyone outside Turkey who has been convinced by the Turkish government."
And Lower Saxony Interior Minister Boris Pistorius went further, saying, "We have to say very clearly that it involves a fear of conspiracy you can class as paranoid."
Germany's foreign secret service have said they will not be commenting but the country's internal security agency is looking into the allegations.
Last week, Swiss prosecutors said they were investigating allegations that Turks critical of the Erdogan government were spied on at a lecture in Zurich.