In his first televised address since the military recaptured the capital Khartoum this week, Burhan said that an end to nearly two years of devastating fighting is possible "if this militia lays down its arms".
He ruled out any negotiations with the paramilitaries, saying that victory will only be complete once "the last rebel has been eradicated from the last corner of Sudan".
Burhan's speech came just days after he triumphantly entered the presidential palace, which had been under RSF control since the war erupted nearly two years ago.
The army's recapture of Khartoum has forced the paramilitaries to regroup but the RSF leadership has continued to voice defiance, vowing "no retreat and no surrender".
Hours after Burhan walked back into the presidential palace, the RSF announced a "military alliance" with a faction of the rebel Sudan People's Liberation Movement-North (SPLM-N), which controls parts of South Kordofan and Blue Nile states.
The war has devastated Sudan, killing tens of thousands and displacing more than 12 million.
The country is now effectively split in two, with the army holding the north and east, while the RSF controls most of Darfur in the west and much of the south.
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