Both Israel and Gaza's Hamas rulers have denied talk of having reached an official cease-fire.
Referring to an understanding between the sides, Aboul Gheit said: "An agreement could include an exchange of prisoners and a lifting of restrictions at Gaza's Israeli-controlled crossings."
The foreign minister added: "If we would not manage to allow that situation to come to an end, then the situation will worsen again and that is not good for the people of Israel or the Palestinians."
In recent weeks, increased rocket fire from the Hamas-controlled Gaza Strip into Israel sparked a series of Israel Defense Forces raids that killed more than 120 Palestinians. Last week, an Israeli Arab terrorist gunned down students at a Jerusalem yeshiva, killing eight.
Egypt's intelligence chief Omar Suleiman on Thursday offered Palestinian factions an Israeli commitment it will cease its military operations against militants in Gaza if the factions made a commitment to halt the rocket fire from the Strip into Israel, the London-based Arabic language newspaper Al-Hayat reported Saturday.
Meanwhile, a rocket fired from the Gaza Strip struck southern Ashkelon on Tuesday, ending a four-day lull in violence that began directly following the terrorist attack at the Mercaz Harav yeshiva in Jerusalem last Thursday.
No damage or injuries were reported.
The rocket hit south Ashkelon only a few hours after Prime Minister Ehud Olmert toured the rocket-plagued city, warning that there is no way of assuring that the rocket fire on the city would not resume in the future.
The Popular Front for the Liberation of Palestine (PFLP) took responsibility for the rocket attack. But Israel held Hamas responsible.
"Hamas controls Gaza and they are responsible for every missile fired from Gaza into Israel. We have no illusions as to the extreme and hateful agenda of Hamas," said government spokesman Mark Regev.
Olmert said during his visit that the rocket fire on Ashkelon could not be seen as an isolated incident, adding that the situation in Ashkelon could not be compared to Sderot.
Abbas to Jordanian reporters: Barak is sabotaging peace process
Palestinian President Mahmoud Abbas told Jordanian reporters in Amman Tuesday that "a top Israeli official is sabotaging the peace talks with Israel over internal matters and due to a personal hostility toward me."
One of the Jordanian reporters who broke the story said that Abbas was referring to Defense Minister Ehud Barak.
Abbas added that talks with Israel regarding the Palestinian refugees' right of return will go on for 10 more years at least.
On Monday, Abbas contradicted Barak's earlier statement and said that a ceasefire agreement had been reached between Israel and Hamas. Abbas' statement came only a few hours after both Barak and Olmert said that no such cease fire had been agreed upon.
Abbas also said Monday that Hamas leaders wanted to reach a ceasefire because they feared for their lives.
Hamas spokesman Sami Abu Zuhri denied Abbas' claims on Tuesday, saying that "these remarks are nothing but lies aimed at damaging Hamas' image."
"Hamas leaders seek martyrdom and would never bargain over the blood of their people like others do," Abu Zuhri said.