The efforts to halt the bloodshed will be the main subject in a meeting next week between Mahmoud Abbas, the Palestinian president, and Ehud Olmert, Israel's prime minister.
Abbas, the leader of Fatah, has been trying to press coalition partner Hamas to renew the truce.
Palestinian killed
Hamas says it has not launched any rockets since Wednesday.
The Islamic Jihad fighter group and others took responsibility for the firing of at least three rockets towards Israel on Saturday. No injuries were, however, reported.
In a separate development in the West Bank, Israeli troops shot and killed a Palestinian and injured three others in the city of Nablus on Saturday.
The Israeli army said troops fired at gunman during an arrest operation in the city.
Truce talks
In an interview with the Egyptian daily Al-Ahram, Moussa Abu Marzouk, Hamas' deputy political leader, said: "We may agree to a one-year ceasefire. Both parties have to abide by it."
Abu Marzouk, who along with Hamas leader Khaled Mashaal lives in exile in Syria, demanded that Israel also agree to a truce with Hamas, saying the only way for success was calm on both sides.
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"Hamas is trying to draw us [Israel] into their civil war"
Zafon, Beersheva, Israel
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Several senior Hamas leaders have been in Cairo over the past week for talks with Egyptian officials on ways to end the fighting with Israel.
Mohammad Nazal, a member of the Hamas political bureau, also confirmed that Hamas is considering a truce.
Nazal said by telephone from Damascus: "Some private ideas were presented to Hamas [by Egyptian mediators] to reach a truce with Israel, and Hamas is about to undertake the suitable decision."
Israel wary
Israel said it was skeptical of Hamas' intentions.
Mark Regev, a foreign ministry spokesman, said: "The current ceasefire in Gaza is unfortunately a sham. It would seem that before we talk about expanding the ceasefire, we should first get it right."
Hamas wants to get Israel to extend any Gaza ceasefire to the West Bank, where Israeli troops frequently conduct arrest raids against fighters.
Under a plan being worked out by Abbas, Gaza fighters would halt rocket fire for a month to allow for negotiations on a more comprehensive ceasefire that would include the West Bank.