This will be through an international crossing at Rafah, whose target opening date is November 25th.
Mr Marzuka said Egypt's contact with Palestinians in Gaza should be through the Rafah crossing.
The Rafah crossing has been closed since 2007, although special medical cases are occasionally allowed through.
During Sunday's Cabinet meeting, Barak too voiced optimism that Egypt would regain control of the Rafah border crossing.
After Israel's withdrawal in 2005, an EU team monitored the crossing at Rafah, on Gaza's southern border with Egypt.
But as diplomats continue their negotiations the worst casualties continue to arrive at Rafah, including some children with appalling injuries.
Correction 7 January 2013: This story has been amended to clarify the reference to Egypt's closure of the Rafah crossing.
Meanwhile, Palestinian officials say the weekend's clashes in the southern Gaza town of Rafah have left at least three Palestinians dead.
Rafah is the crossing point on the border between Egypt and Gaza.
The Rafah crossing disturbance involved 100 policemen, who staged a sit-in within the terminal to protest the shooting death of a policeman Thursday.
As the search for any remaining gunmen continues, Egypt has closed its Rafah border crossing with Gaza, our Cairo correspondent Yolande Knell reports.
And on Friday it partially reopened the Rafah border crossing between Gaza and Egypt, which had been closed since an attack in early December.
They've gotten some surveillance cameras, I guess, at the Rafah crossing.
Early on Tuesday he was taken to the Rafah crossing between Gaza and Egypt, and handed over to Egyptian mediators by Hamas, in the presence of Israeli representatives.
BBC: Gilad Shalit freed in Israeli-Palestinian prisoner swap
Meanwhile, at the Gaza-Egypt border, nearly 25 trucks carrying aid and medical supplies were unable to pass through the Rafah border crossing on Sunday, CNN's Karl Penhaul reported.
Egypt's military leadership, the Supreme Council of the Armed Forces, said 35 assailants -- terrorist factions from Gaza -- were involved in Sunday's attack near the Rafah border crossing.
Rafah is also the area where Kate Burton, a 25-year-old British worker for the Palestinian rights group Al Mezan, was kidnapped with her parents while showing them around Wednesday afternoon.
The Rafah crossing was open sporadically between June and January, when Egypt ordered it opened to those in need of medical care, students, and foreign passport and residency card holders.
It has been conducting trials with 1, 500 machines in the region and has begun to distribute a further 2, 100 to a school in Rafah, in the south of the Gaza strip.
Jabari therefore shocked many observers when he appeared on television personally escorting Sgt Shalit to the Rafah border crossing with Egypt in October 2011, after a prisoner exchange was agreed between Hamas and Israel.
Israel, meanwhile, has remained notably silent about Egypt's decision last week to open the crossing between Gaza and Sinai at Rafah and its follow-up announcement on June 7th that the border would remain open indefinitely.
Wrongly assuming that the Europeans shared Israel's interest in preventing terrorists and weapons from entering Gaza, Israel requested that EU monitors set up shop at the Rafah terminal linking Gaza to Egypt after Israel withdrew from the border in 2005.
More than a dozen Egyptian soldiers were killed and seven others were wounded in an August 5 attack near the Rafah border crossing, when assailants with semiautomatic weapons and hand grenades stole two armored vehicles from Egyptian forces and tried to enter Israel.
The raid comes a week after 16 Egyptian soldiers were killed and seven others were wounded in an attack near the Rafah border crossing, when assailants with semiautomatic weapons and hand grenades stole two armored vehicles from Egyptian forces and tried to enter Israel.
Such has been the experience, for example, with United Nations forces in southern Lebanon, European Union monitors who were deployed for a time at the Rafah crossing between Egypt and Gaza and even U.S.-led multinational forces deployed to monitor the demilitarization of the Sinai.
Officials in Egypt long maintained that they shut their Rafah crossing with Gaza (where, incidentally Sgt Shalit was freed this morning) on the request of the PLO, which blamed Hamas for killing large numbers of supporters from President Mahmoud Abbas' Fatah party during their takeover.
This time I was worried that something similar would happen so I prepared the paperwork to cross into Gaza from Rafah (on the Egyptian side) but also decided to gamble on taking an early flight to Tel Aviv just in case I needed to cross in from Israel.
At the start, America tried to keep the gates open, brokering an Agreement on Movement and Access with Israel to allow the export from Gaza of hundreds of trucks of produce a day, regular bus convoys to and from the West Bank and the opening of a Palestinian-controlled crossing at Rafah to Egypt.
Some in Israel's security establishment have privately expressed concerns that the increased traffic at Rafah could serve to allow more militants and weapons to cross in and out of Gaza and that it could ultimately serve to bolster the position of Hamas, which Israel and the United States consider a terrorist organization, but the Israeli government has said little publicly.
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