Abu Marzouk suggested that Abbas is wasting his time at the U.N. "Hamas believes the General Assembly is not the one to create states," he said. "Occupation needs resistance, not negotiations."
Israel and the West have shunned Hamas, which has killed hundreds of Israelis in shootings and bombings over the years, as a terror organization. However, Hamas officials believe the boycott is slowly eroding, pointing to U.S. support for the cease-fire deal brokered by Egypt and the ongoing indirect negotiations between Israel and Hamas.
Overall, Hamas leaders have claimed the group has emerged victorious from this round, noting that Israel did not make good on threats to send ground troops into Gaza. Israel says it has achieved its goal of halting rocket fire on Israel.
Abu Marzouk said the next round of indirect talks will take place in Cairo on Monday. He has not met his Israeli interlocutors, he said, but said they are security officials and experts on border arrangements.
Until late last year, most top Hamas leaders in exile were based in Syria, the Islamists' main foreign backer in addition to Iran. However, Syrian President Bashar Assad's brutal crackdown on a popular uprising there made Hamas' alliance with the Damascus regime untenable.
Abu Marzouk, who has settled in a quiet Cairo suburb, said the follow-up talks with Israel were going well so far.
In Gaza, residents said Saturday that Israel has already eased some restrictions.
Fishermen were able to sail six nautical miles out to sea, or double the previous limit, said Mahfouz Kabariti, head of the local fishermen's association. "This is an opportunity and a chance for a better catch, though it is still a limited area," said Kabariti, who represents some 3,500 fishermen.
Israeli navy boats have been enforcing a sea blockade in an attempt to prevent weapons smuggling to Gaza. The restrictions on fishermen have fluctuated over the years, linked to the ups and downs in Israeli-Palestinian relations.
Meanwhile, some Gaza residents said they were able to enter an Israeli-enforced buffer zone on the Gaza side of the border Saturday with Israel without fear of being fired on.
Israel's military had carved out a 300-meter-wide (300-yard-wide) zone several years to try to prevent militants from sneaking into Israel. The zone gobbled up scarce farmland in one of the most densely populated areas in the world.
On Saturday, 42-year-old farmer Nidal Abu Dakka said soldiers stood and watched as he and others moved close to the fence. In other border areas, residents said Hamas police kept them away from the fence.
An Israeli government spokesman said he was unaware restrictions had been eased. A defense official said the Israeli military was no longer enforcing the no-go zone, but reserved the right to act against suspicious people. Both spoke on condition of anonymity because they were not allowed to discuss the issue with reporters.
El Deeb reported from Gaza City. Associated Press writer Karin Laub in Gaza City contributed reporting.