The Tigers fought for two decades against the Sri Lankan government in a war that left more than 64,000 dead until a ceasefire in 2002 left them running their own courts, bank, law school, police, customs department and tax system in areas they control.
"Those who engage themselves in writing the Tamil national anthem should give life and energy with great importance to express the greatness, the glory and the resources of the Tamil homeland," the Tigers said in an appeal on Friday.
"It should contain 18 lines - catchy and lively and in pure Tamil."
The Tigers, who say they want contributions from Tamils both in Sri Lanka and abroad, circulated the statement to media offices in the government-controlled north and east, including the largest northern town, Jaffna, as well as on pro-rebel website www.tamilnet.com.
The Tigers currently have no anthem, usually falling back on the tune played when their flag is raised - entitled Look the Flag is Rising - but the group said it believed a national anthem had become vital, although it did not say why.
"The national anthem should include the price paid by our generation towards the achievement of our homeland so that the words should live forever in the memory of future generations," the statement said.