The figure is a total for the period August 2016 to January 2017, German media report.
Turkey has urged Germany not to grant asylum to any military officers. Some posted to Nato bases in Germany are thought to be among the group.
In Greece, two more Turkish soldiers have requested asylum.
The pair - reported to be commandos - are believed to have taken part in the failed coup against President Recep Tayyip Erdogan.
They are in Greek police custody, having applied for asylum last week in Orestiada, a small border town near Turkey.
Last month a Greek court rejected Turkey's request to extradite eight other Turkish soldiers who fled after the coup attempt. Turkey is appealing against that ruling.
The German interior ministry did not identify the 136 Turks who requested asylum. Not only diplomats but also their spouses and children hold diplomatic passports.
It is not clear if any of them have been granted asylum yet.
Soldiers who fled after the coup attempt fear that they will not get a fair trial in Turkey.
The Turkish authorities have dismissed at least 100,000 public servants, including teachers, police and members of the judiciary.
About 43,000 suspects are in detention. The crackdown is targeting suspected supporters of Fethullah Gulen, a cleric who lives in self-imposed exile in the US.
Mr Erdogan's ruling AK Party has also shut down more than 130 media outlets, during a continuing state of emergency, and arrested about 150 journalists.
Several prominent German politicians have spoken out against a possible campaign visit by Mr Erdogan. Turks will vote in a referendum on 16 April to decide whether Mr Erdogan should have sweeping new presidential powers.
The referendum will ask Turks if they agree to constitutional changes to make Turkey a presidential state, reducing parliament's powers.
Turkish Prime Minister Binali Yildirim addressed a rally in Oberhausen, north-western Germany, on Saturday and said Mr Erdogan had plans to visit the region in March. But it has not been officially confirmed.
About 1.4 million Turks living in Germany are on Turkey's electoral register.
Oberhausen is in the state of North Rhine-Westphalia, where state interior minister Ralf Jaeger warned that "we must prevent Turkish internal conflicts being played out here".
The Social Democrat (SPD) politician said Turkey's controversial referendum would lead to "restricted fundamental rights and a return of the death penalty".
German-Turkish relations have been severely strained by the AKP's post-coup crackdown.
Recently German police searched the homes of four imams amid reports that Ditib, a Muslim organisation funded by the Turkish state, was spying on suspected Gulen supporters in Germany.
Ditib is short for the Turkish-Islamic Union for Religious Affairs.
North Rhine-Westphalia is also investigating claims that Turkish diplomats asked school pupils and teachers in some schools to report any criticism of Mr Erdogan to them.
Turkey has accused German officials of conducting a "witch-hunt", telling them to focus instead on "terror organisations" such as the Kurdistan Workers' Party (PKK).
Political tensions are already running high in Germany ahead of crucial parliamentary elections in September.