The magazine is offering £1,000 ($1,450), donated by a reader.
It comes after German Chancellor Angela Merkel said Jan Boehmermann could be prosecuted over a TV broadcast.
Jan Boehmermann had recited a satirical poem on the TV channel ZDF which made inappropriate references to Recep Tayyip Erdogan.
The comedian is now under police protection and Angela Merkel’s government has approved a criminal inquiry, under a little-used law concerning insults against foreign heads of state.
Angela Merkel stressed that the courts would have the final word, and it was now up to prosecutors to decide whether to press charges.
Announcing the competition, commentator Douglas Murray wrote: “The fact such a trial could even be contemplated demonstrates that Germany is becoming little more than a satrapy [province] of Erdogan’s.”
“I’m a free-born British man… In honor of this fact I have spent the weekend writing rude limericks about Mr. Erdogan.
“And I would hereby like to invite all readers to join me in a grand Erdogan limerick competition.”
Since Recep Tayyip Erdogan became president of Turkey in 2014, almost 2,000 cases of insulting him have been opened.
Jan Boehmermann case has opened a debate about free speech in Germany.
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