During a Saturday demonstration in the Swiss capital, organised by Kurdish groups, some people brandished a banner showing a gun pointing at the Turkish president alongside the words “Kill Erdogan”.
Turkey summoned the Swiss ambassador on Sunday to protest at the rally, claiming it was organized by the Kurdistan Workers' Party (PKK), which Ankara says is a terrorist group.
An investigation targeting unidentified people over possible incitement to violence has been opened by cantonal authorities in Bern, Switzerland's ATS news agency said.
Prosecutors in Istanbul have opened a separate investigation, accusing unnamed suspects of “membership in a terrorist organization” and “insulting the president”, Turkey's state-run Anadolu news agency said.
The two probes come amid increased tensions between Bern and Ankara and with Erdogan's administration launching regular rhetorical attacks on Europe as a whole.
Turkey has been fuming since its government ministers were blocked from holding campaign rallies in several European countries to support next month's referendum on expanding Erdogan's powers.
The Swiss federal government had approved a campaign rally by Turkish foreign minister Mevlut Cavusoglu, but it was called off after the hotel booked for the event cancelled.
Erdogan on Sunday criticized the Swiss for hosting the Kurdish rally, warning that “you will reap what you have sown”.
Europe was “living through an age of ignorance”, he told a youth rally in Istanbul on Monday, according to televised remarks.
Turkish Prime Minister Binali Yildirim on Monday called the Swiss rally banner “unacceptable” and branded it “a crime”, according to remarks broadcast on the private NTV television station.
Meanwhile, Swiss prosecutors opened an investigation last week into alleged spying on Turks living in Switzerland by an unspecified intelligence service after the attorney general's office cited “concrete suspicions” of espionage.