It is the latest case in which youngsters have been hauled before the courts in Turkey on similar charges, fuelling concerns about freedom of speech under Erdogan.
Turkish teen detained for 'insulting' Erdogan
The prosecutor in the main Kurdish city of Diyarbakir is seeking jail terms of between 14 months and four years and eight months for the pair, according to the Hurriyet newspaper.
It said the two were caught in May ripping down the Erdogan poster on a Diyarbakir street but defended themselves saying they just wanted to sell the paper.
"We did not pay attention to who was in the picture, we do not know who it was," the younger defendant, identified only by the initials K.Y., told a magistrate.
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"If the president is prosecuting two children who tore down a poster, it's very sad for justice," said their lawyer Ismail Korkmaz.
The boys are being prosecuted under an article of the penal code that punishes anyone who "undermines the image" of the head of state with up to four years in prison.
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Critics say Erdogan has become increasingly authoritarian since becoming president in August 2014 after more than a decade as prime minister and the number of prosecutions for insulting him have risen.
Artists, journalists and schoolchildren have all been targeted.
Last December, a minor aged 17 was arrested in his classroom and held for several days on the same charge and given an 11-month suspended sentence, triggering outrage.
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