"We hope that we can reach a satisfactory solution for us, because it's important to have full freedom for people to come in and out," Abbas said on Sunday.
"This subject is under serious discussion, but there is no agreement at this time ... We hope that we can reach solutions for all the crossings very soon."
Abbas spoke after a meeting with international Middle East envoy James Wolfensohn, who has been trying to broker an agreement over the border crossing and other economic issues surrounding the Gaza pullout.
The crossing into Egypt is the Palestinians' most important link to the outside world. The Palestinians say free movement through the border will be essential for reviving Gaza's economy after a 38-year Israeli occupation.
Israel wants to maintain some control over the border, fearing weapons could be smuggled into Gaza.
Wolfensohn said he would meet Israeli officials this week to discuss the dispute.
Israel has said it plans on constructing a new border terminal at the point where Gaza, Egypt and Israel meet, near the Israeli communal farm of Kerem Shalom.
Israel wants all goods and people flowing into Gaza to go through the new terminal and be inspected by Israeli personnel.
The Palestinians say they are willing to route imports through Kerem Shalom to preserve a customs union with Israel, but want exports and people to go through the Rafah terminal inside Gaza, with inspections by third party monitors rather than Israelis.
Palestinians say without a deal
the occupation will not end
Palestinians say that without an agreement on that and other issues - including the opening of a harbour and airport - Israel's 38-year-old occupation of Gaza would not be really ending.
After Sunday's meeting, Wolfensohn said the two sides were close to a deal on the removal of rubble from Jewish settlements.
Pullout almost complete
Israel has evacuated all 21 settlements in Gaza and last week completed demolition of the homes left behind.
Under an emerging deal, Israel will pay for the removal of hazardous materials, while the Palestinians will use recyclable debris for construction projects.
Wolfensohn said the Israelis agreed to pay for clearing the debris - expected to be $30 million - and that he had been meeting private Gaza businessmen to discuss uses for the rubble.
Israel's army chief, Lieutenant-General Dan Halutz, said on Sunday that he expected to complete the Gaza pullout around 15 September.
Israel has destroyed almost all
of its settlements in Gaza
Aljazeera's correspondent in Palestine, Walid al-Omari, confirmed the pullout was close to completion.
"The Israeli army has fully destroyed settlement buildings in Gaza after the evacuation with the exception of the Jewish synagogues," he said.
This stage of the pullout included transferring remains from cemeteries, he said, adding: "The army finished dismantling and transferring its military installations and equipment."
As part of an agreement with Egypt, Israel will withdraw its troops from a patrol road along the Gaza-Egypt border, and 750 Egyptian troops will deploy there to prevent arms smuggling into Gaza.
The Israeli Defence Ministry said on Sunday that the Egyptians would begin taking up positions along the border in the coming days.
"We hope that we can reach a satisfactory solution for us, because it's important to have full freedom for people to come in and out"
The forces are to patrol along a volatile corridor on the Gaza side of the border where Israeli-Palestinian fighting has been intense.
Halutz said he expected the forces to be effective in stopping gun-running.
"It is in the Egyptians' interest to prove that the border is blocked to smuggling," he told reporters. "And I assume they will carry out this obligation."
Israel and Egypt signed the agreement last week, overriding a demilitarisation clause in the 1979 peace treaty between the two countries.
Meanwhile, Palestinian Interior Minister Major-General Nasr Yousif said no group other than the Palestinian Authority had a right to bear arms in the streets.
Yousif was speaking after a meeting between Abbas and members of Fatah movement central committee in Gaza.