Buschmann told the Bild am Sonntag that the Emirati-based messenger had to have a contact person in Germany and a way of flagging and taking down what Germany sees as illegal content.
If Telegram fails to respond, he said, "the next step will be a public notice. Then we can impose fines. Our laws allow for millions worth of fines."
Buschmann promised to take a tough line on threats against police and health officials made online, including on Telegram, after strict COVID-19 rules prompted protests. More than 13,000 rallied against restrictions across Germany on Saturday.