The draft agreement envisages Abdullah leading the peace process with the insurgent Taliban while Ghani leads the country as president, senior sources within both camps said.
Discussions over some key posts were still underway but if an agreement was struck, the deal could be signed as early as Sunday afternoon, the sources told Reuters.
"Overall both teams have reached an agreement based on which they will have 50 percent share in the government. The thing that is still under debate right now is about which ministry should go to whom," said one of the sources.
Abdullah wants to control a major portfolio such as finance or foreign affairs and while Ghani has not agreed to this, he could offer control of the interior ministry, sources said.
Abdullah had disputed the results of the September election and announced the formation of a parallel government earlier this year, undermining Ghani's administration at a time when the United States was trying to advance a peace process with the Taliban.
Washington, frustrated by the growing impasse between the two men, even after Secretary of State Mike Pompeo travelled to Kabul to mediate, threatened a $1 billion aid cut if the men could not agree.
It was not immediately clear whether the agreement, if reached, would result in aid being reinstated, but it comes as Afghanistan faces growing fiscal pressures, with tax revenues falling and foreign aid pledges due this year expected to shrink.
Officials say that a deal between the Ghani and Abdullah is crucial to launching peace talks, as Abdullah's camp represents much of the country's north-west.
But the talks face a number of stark challenges, as violence in the country increases and a harrowing attack on a Kabul maternity ward this week prompted Ghani to switch the military to an "offensive" stance against insurgent groups.
The Taliban deny involvement in the attack, but the government has remained skeptical and angry at ongoing Taliban attacks against the Afghan military, fraying the momentum for peace talks, which were due to start in March.
US Special Envoy Zalmay Khalilzad said Friday that a new date for intra-Afghan peace talks was under discussion and he would soon travel to the region and try to encourage a reduction in violence.