Bashir, who is wanted by the international court for crimes against humanity, said he would recognize the independence of South Sudan once the region becomes the world's newest nation in July.
Nearly all of the voters who took part in a secession referendum voted in favor of the measure, which was part of a comprehensive cease-fire brokered in 2005.
Rabi Abd al-Ati, a ranking member in the ruling National Congress Party, said Bashir would step aside in 2015.
"President al-Bashir will not run in the next election in four years," he was quoted by CNN as saying. "He will also leave his post as chair of the NCP to allow for the transformation of power to a new generation."
He denied, however, the Bashir's decision was a reaction to the wave of political unrest sweeping across much of the Middle East and North Africa.
Bashir took power in a coup in 1989. He won re-election last year in a vote largely boycotted by his opponents.
Ati said Bashir didn't fear international prosecution because Sudan wasn't a party to the International Criminal Court.
Toddler uninjured after being knocked over by Obama family dog
Gal Gadot cast as Wonder Woman for 'Batman vs. Superman'