According to the sources, Israel and Hamas have agreed to the Egyptian proposal to deploy Palestinian Authority Presidential Guard members along the Karni, Sufa and Kerem Shalom crossings, where cargo is transferred between Israel and Gaza, as well as at the Erez crossing, a passageway for people and goods.
Guard members will be deployed also at the Rafah crossing, which connects Sinai, Egypt with the Strip.
Hamas forces will be positioned nearby and will essentially control the movement of Palestinian civilians in and out of Gaza.
This agreement is in keeping with the 2005 crossings agreement, between the U.S., Israel and the PA, which called for placing the crossings under forces loyal to Abbas.
The Palestinian sources said that Hamas leaders are due to meet with Egyptian mediators within the next two days. The Egyptians will convey Israel's views on the Egyptian proposal, and Hamas is to state whether it agrees to a temporary cease-fire.
The current round of violence ended after Hamas and Islamic Jihad leaders met with Egyptian officials last week.
The PA also has agreed to the Egyptian proposal. It is expected to present any agreement as a success, because this would mean Hamas has failed to break the siege on Gaza. It has controlled the coastal strip since overthrowing Fatah there last June.
Meanwhile, Damascus-based Hamas leader Mohammed Nasser released the group's conditions for a cease-fire with Israel Tuesday. Nasser says Hamas is demanding Israel completely cease "all acts of aggression" in the West Bank and the Gaza Strip.
By this demand, Hamas is toughening its stance regarding the cease-fire in the works, and is posing a demand that Israel steadfastly has refused to accept so far. Israel rejects any deal that would bar it from arresting suspected militants in the West Bank.
Ayman Taha, a Hamas spokesman in the Strip, confirmed in a telephone conversation with Haaretz on Tuesday that the radical Islamic group is demanding Israel stop making arrests in the West Bank in exchange for a cease-fire deal.
Nasser also added that Hamas is ready for a temporary cease-fire with Israel, on the condition that it occurs simultaneously and applies to both sides.
Abbas said Monday during a meeting with Jordanian journalists that "a senior figure in the Israeli government is undermining the negotiations for internal reasons and because of personal hostility to me." He was in Jordan to meet with King Abdullah.
A Jordanian journalist who published the story noted that Abbas was referring to Defense Minister Ehud Barak.
Abbas also said that any agreement on Palestinian refugees' right of return would be implemented over the course of at least a decade after the signing of a peace deal.