Mediators, including the US, reportedly expressed support for Barghouti's release.
Yahya Sinwar discussed the proposed deal through Hamas representatives for the first time on Friday, saying it was the closest offer yet to the terrorist organization's demands.
The immediate release of terrorists incarcerated by Israel has remained a repeated demand by Hamas and recently, it added Barghouti to the list of those who must be freed.
Marwan Barghouti, the head of Tanzim, an armed faction of Fatah, was arrested by the Israeli security forces in the West Bank city of Ramallah on April 15, 2002.