Gen Gilbert Diendere made the proposal at talks brokered by West African mediators in the capital Ouagadougou.
On Saturday one of the mediators had spoken of a breakthrough and hinted at a new transition government reinstating interim President Michel Kafando.
At least 10 people have been killed in clashes since Thursday’s coup.
The overthrow of the civilian interim government – carried out by the presidential guard – was widely condemned. Burkina Faso has been suspended from the African Union.
BBC West Africa reporter Thomas Fessy says the junta’s proposal is unlikely to signal a return to power of the civilian authorities.
The document, signed by Gen Diendere and exclusively seen by the BBC, says he should remain president until elections – currently due on 11 October.
This is in stark contradiction to the optimism shown on Saturday by one of the mediators, Benin President Yayi Boni, who suggested the presidential guard might give up power, our correspondent adds.
Gen Diendere was chief of staff of former President Blaise Compaore, who was deposed in a popular uprising last October.
Meanwhile violence erupted on Sunday at the hotel in Ouagadougou, where the talks have been taking place.
About 50 coup supporters burst into the lobby of the Laico hotel, injuring several people.
“They invaded the hotel,” an eyewitness told Reuters news agency. “They attacked ex-opposition members as they arrived. One had to be saved from the crowd by security forces.”
Outside the building opponents of the coup held a protest but were later dispersed by security forces.
Mr Kafando, who was initially held by the coup leaders, is now free. However, other questions remain, including the fate of Prime Minister Isaac Zida, who was also detained.
Mr Compaore is currently in exile and was accused of committing widespread abuses, and trying to change the constitution to extend his term in office.
Some of his key allies had been barred from contesting the election.
Gen Diendere has said he has had no contact with Mr Compaore and will do everything to “avoid violence that could plunge the country into chaos”.