Egypt has vowed to regain control of its border with the enclave.
As extra security arrives and shops close there, diplomatic efforts have been under way in Cairo and Jerusalem.
Both Hamas and Fatah have accepted invitations to hold separate talks with the Egyptian government to discuss the situation.
Ehud Olmert, the Israeli prime minister, and Mahmoud Abbas, the Palestinian president, held crisis talks on Sunday in Jerusalem on the border breach.
An Israeli official said: "[Olmert] reassured Abbas that Israel would continue to allow humanitarian aid into Gaza after Abu Mazen [Abbas] requested that Israel not harm the civilian population in the Gaza Strip."
|Palestinians in Gaza have suffered shortages|
of power, fuel and basic goods [AFP]
He said Israel and Egypt were also in talks on how to reseal the border with Gaza.
Abbas reportedly wants to take over Gaza's border crossings, including the one at Rafah.
Saeb Erekat, a Palestinian negotiator, said on Sunday that Abbas is ready to shoulder the responsibility.
Israel has so far resisted the idea, citing concerns about security.
Al Jazeera's correspondent said that the Olmert-Abbas meeting did not address the Palestinian Authority's [PA] offer to assume the management of the Gaza crossings.
Erekat further cautioned Israel against shifting responsibility for Gaza on to Egypt.
"Israel is fully responsible and any games to try to divert this responsibility to Egypt is absolutely unacceptable," he said.
Erekat said Israel is attempting to separate the West Bank from the Gaza Strip.
In Cairo, Ahmed Aboul Gheit, Egypt's foreign minister, said on Sunday after a meeting with Salam Fayyad, the Palestinian prime minister, that "Egypt will take necessary actions and measures to control the border in Gaza soon".
For his part, Fayyad said: "I believe there is a need, and also a consensus, to support the trend backed by the international community, that is to reopen the border crossings and to restore their administration to the Palestinian Authority.
"This will be a step to enable us to end not only the suffering, but also the siege imposed on Gaza Strip."
Sami Abu Zuhri, a Hamas spokesman, said: "We are interested in offering the alternative and that is the opening of the Rafah border crossing, and for that we call upon the Egyptian leaders to take an urgent and quick decision regarding the subject matter."
Until the Hamas takeover of Gaza in June last year, the Rafah terminal was run jointly by Egypt, PA security forces and European observers, with surveillance cameras allowing Israel to monitor those passing through.
Meanwhile, Al Jazeera reported that Khaled Meshaal, the group's political bureau chief, arrived in the Saudi capital Riyadh on Sunday at the head of a delegation.
The visit came as Arab foreign ministers kicked off a meeting in Cairo to discuss the situations in Gaza and Lebanon, and Israel launched two air strikes against Hamas targets near Rafah without causing casualties, according to witnesses.
In related news, Israel may be obliged to resume fuel shipments to the Gaza Strip.
The move was approved by the Israeli supreme court on Sunday as it heard a case regarding the legality of the siege.
, the Israeli daily, said the country's highest court held a hearing to discuss a petition filed by several human-rights organisations against the reduction in Israeli fuel and gas shipments to Gaza.
|Palestinians have been moving freely between|
Gaza and Egypt since Wednesday [EPA]
However, the government previously contested the resumption of fuel.
It said that the levels soon to be shipped into Gaza are "the minimum required in order to meet the basic humanitarian needs of the strip's civilian population".
People in Gaza have suffered shortages of power, fuel and basic goods.
Al Jazeera's David Chater, reporting from Jerusalem, said the decision means that the immense pressure placed on the Palestinians in Gaza will be reduced.
He said: "This will also have an effect on the developments in the Rafah crossing."
Al Jazeera's Jacky Rowland, reporting from the Gaza side of the Rafah crossing, said that the Egyptian government is using economic means to restrict the flow of Palestinians across the border.
She said: "Shops in Rafah, as well as in the town of El-Arish, 50km from Rafah, have been ordered to close for business.
"However, as we have seen for the past few days, the sense of freedom felt by the Palestinians is something they will not easily give up."
Hamas armed forces on Sunday set up checkpoints in Rafah, preventing cars from Gaza from entering Egypt, a correspondent of the AFP news agency said.
Pedestrians were still allowed to cross freely, although persistent rain during the day reduced numbers.
Meanwhile, Egyptian forces moved to close the border by stopping vehicles and further tightening their security cordon around the town of Rafah.
Security guards blocked one of the gaps carved into the border wall with piles of sand and border police were stopping cars with Palestinian number plates from entering Egypt and Egyptians cars from crossing into Gaza.
Separately, dozens of Egyptian lorries loaded with goods were blocked on Sunday from crossing into the Sinai.
The vehicles, ranging from small pickup trucks to large lorries, lined the road leading up to the Mubarak peace bridge which crosses the Suez Canal into the Sinai.