During Saturday's demonstration in the capital Bern, organised by Kurdish groups, some people brandished a banner showing a gun pointing at Erdogan alongside the words "Kill Erdogan".
Turkey summoned the Swiss ambassador on Sunday to protest at the rally, claiming it was organised 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, ATS said.
Erdogan criticised the Swiss for hosting the event, warning that "you will reap what you have sown".
Relations between Ankara and Bern have grown increasingly complex in recent days.
Cantonal authorities in Zurich also tried to block Turkish Foreign Minister Mevlut Cavusoglu from campaigning earlier this month in support of next month's referendum to expand Erdogan's powers.
The Swiss federal government said the campaign events could go ahead, although Turkish officials called off the rally after the hotel booked for the event cancelled, and after other European states blocked similar events.
(Except for the headline, this story has not been edited by NDTV staff and is published from a syndicated feed.)
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