A delegation led by Khaled Meshaal, the leader of Hamas, has proposed joint Palestinian-Egyptian control of the Rafah crossing.
They set down these conditions on Thursday on the second day of talks in Cairo geared towards resolving the week-old crisis on the Gaza-Egypt border as Egypt continued its mediating role between Hamas and Fatah.
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The delegation proposed removing EU observers, maintaining that these observers were in no position to stop Israel from closing the border crossing.
Mahmoud al-Zahar, a Hamas leader, said these "historical borders" should not be used to impose a seige on the Palestinians.
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Other conditions set by Hamas include that the Palestinian role should be determined by an agreement between Hamas and the Palestinian Authority (PA).
The movement also said it was against Israel's veto power to close the crossing.
Hamas further proposed that the entry into and out of Gaza through Rafah should not only be confined to Palestinians carrying identification cards issued by the PA.
Thursday's deliberations come a day after Mahmoud Abbas, the Palestinian president, met Hosni Mubarak, Egypt's president, and asserted his refusal to deal with Hamas.
Meshaal is expected to meet Ahmed Abulgheit, Egypt's foreign minister, and Omar Sullieman, its head of intelligence.
'Combatant' killed
In related news on Thursday, Israeli soldiers killed a Palestinian man in the southern Gaza Strip, local medics and the Israeli military said.
Twenty-year-old Mahmud al-Daalsah, a member of the Al-Aqsa Martyrs Brigades, was killed in an exchange of fire with Israeli soldiers near the town of Rafah, medics and witnesses said.
An Israeli military spokesman said "the combatant was killed when he approached the security barrier" between the territory and Israel.
Since Israel and the Palestinians revived their peace negotiations in late November after a seven-year freeze, at least 145 people, mostly fighters, have been killed by Israeli troops in Hamas-run Gaza.
Thursday's death brings to 6,105 the number of people killed, a vast majority of them Palestinians, since the start of the second Palestinian uprising in September 2000, according to an AFP tally.
Border closure
Egypt increased security around the border town of Rafah on Tuesday, and resealed parts of the barrier destroyed a week ago by Hamas fighters, in an attempt to control the flow of people in and out of the Gaza Strip.
Egyptian state media has reported Cairo is soon to close the border.
One state-run newspaper said Cairo planned to close the border on Thursday, and the al-Ahram newspaper said the last opportunity for Gazans to return home would be at the start of next week.
The situation for the Gaza Strip, however, remains unresolved, and the Israeli Supreme Court upheld the government's decision to cut fuel and electricity shipments.
Gaza sanctions
Israeli human rights groups have challenged the sanctions, which Israel says are aimed at halting ongoing rocket fire by Gaza fighters.
Palestinian officials say the cutbacks have harmed Gaza's already impoverished residents by causing power shortages and crippling crucial utilities.
Israel, which pulled out of Gaza in 2005 after a 38-year occupation, supplies all of Gaza's fuel and more than two-thirds of its electricity.
The Israeli blockade of the strip, appeared to be easing, however, when Asher Luk, the manager of Israel's Karni crossing with Gaza, said that 70 trucks of wheat and animal feed would be allowed into the Gaza Strip.
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