The Egyptian Chief of Intelligence, General Omar Suleiman, who has been mediating between the parties, is scheduled to meet Tueday with the heads of the Palestinian factions in the Strip.
Israel is waiting to learn from Suleiman whether the Palestinian groups, headed by Hamas, will agree to an unofficial deal on a cessation of terrorist activities in the strip, in return for an end to IDF attacks.
Israel does not intend to officially announce that it has accepted the tahdiyeh deal, but will let the situation unfold gradually - and evaluate the indirect accord with Hamas on the basis of results on the ground.
Tuesday's meeting between Suleiman and a Hamas-led delegation from the Gaza Strip headed by the deputy head of the group's politburo, Mussa Abu Marzouk, is critical to whether the cease-fire deal will be closed.
Suleiman will present Israel's position on the cease-fire and its other demands to the Hamas representatives.
According to Egyptian sources, Hamas will find the Israeli position acceptable.
Suleiman will meet with Israeli officials following the meeting with Hamas, and a decision will be made on H-hour, the point at which the cease-fire will go into effect.
Barak on Monday met with Egyptian President Hosni Mubarak, Suleiman, Foreign Minister Ahmad Aboul Gheit, and Defense Minister Field Marshal Hussein Tantawi.
During his meetings, Barak presented the Egyptians with Israel's conditions for a cease-fire with the Palestinian groups in the Gaza Strip.
Barak broke down the stages of the cease-fire, and said that the IDF will conduct itself within an agreement for "security calm" only after a complete cessation of the Qassam rocket attacks targetting communities in the western Negev, as well as all terrorist attacks originating in the Strip.
Israel is also demanding that at this stage Hamas cease smuggling weapons, funds and persons trained in paramilitary activities.
The Egyptians told Barak that they will step up their efforts to prevent weapons smuggling from Sinai into the Strip, and claimed that they are already seeing a major improvement.
Last year the two sides agreed to set up a hotline for dealing with problems along the border, but this has yet to materialize.
However, the issue of smuggling was the subject of discussion for officers and officials from both sides at a recent meeting.
Egyptian sources said that Israel presented a position that offers the cease-fire agreement real chances of success.
"Both sides, Hamas and Israel, appear interested in the cease-fire," the Egyptian source said.
However, Barak also warned that "if the rocket attacks on the western Negev communities and the terrorist attacks from the Gaza Strip continue, this may accelerate the downward spiral toward a military confrontation between Israel and Hamas in the Strip."
The defense minister insisted that even if there is a cease-fire in the Strip, Israel will retain the freedom of action in the West Bank.
Barak also presented the Egyptians with another demand: "Israel is asking for furthering and accelerating the negotiations for the release of the abducted soldier Gilad Shalit as part of the overall process of settling the situation in the area [the Gaza Strip]."
Egyptian sources said that Israel did not condition the agreement for a cease-fire on the release of the abducted soldier, but added that Israel would like to renew efforts to restart the frozen process of releasing prisoners.
Egypt has made it clear that if the cease-fire agreement is accepted, then the efforts for the release of Shalit will pick up pace.
A senior Hamas source said yesterday that the Egyptians have not tried to link the question of Shalit and the cease-fire. The same source said that the Hamas does not believe Israel will foil the tahdiyeh over Shalit.
"It is a very complicated issue that may delay the cease-fire for weeks," he said.