"It was a harsh meeting but we will be continuing contact and dialogue [with the Israelis] because we are asking for our rights," a Palestinian source quoted Abbas as telling a group of over 100 Palestinian mayors who met to discuss violent fighting in the territories.
"Maybe the meeting didn't achieve important things but we will continue these talks," Abbas said on Saturday.
He also pledged that the recently-postponed legislative elections would happen "very soon", although he did not give a date.
"We are waiting for the Palestinian Legislative Council to finish with everything related to the [new electoral] law," he said.
Last week, deputies cleared the way for holding the legislative elections when they approved proposals for a new electoral system.
"It was a harsh meeting but we will be continuing contact and dialogue [with the Israelis] because we are asking for our rights"
Differences over the voting system earlier this month had forced Palestinian officials to postpone the 17 July elections.
Commenting on the breakdown in law and order spreading through the Palestinian territories, Abbas admitted there were "some problems" but promised to deal with them "in a wise way, to protect people's safety".
Earlier on Saturday, Abbas met a delegation of lawyers who also called for urgent action to curb the wave of violence.
Towns and villages across the West Bank have been held hostage by a rising tide of violent crime, causing alarm among both officials and the public.
Earlier this week, armed men shot dead a 24-year-old policeman in the northern town of Jenin, prompting the Palestinian security forces to launch an arrest campaign.
In another step to defuse the situation, Prime Minister Ahmed Qureia is meeting Islamic Jihad leader Shaikh Abdullah al-Shami on Saturday night to discuss the recent escalation in tension between the Israeli army and the fighters of the group, reported Aljazeera.
Israel had in recent days arrested several members of the Islamic Jihad and warned it would resume assassinations provoking a threat of retaliation from the resistance group.
On Saturday, Israeli troops arrested 14 more Islamic Jihad cadres in Ram Allah, reported Aljazeera.
Settlers to leave
Qureia to meet Islamic Jihad
leader on Saturday night
Meanwhile, the first group of Jewish settlers on Saturday
announced they plan to move out of their Gaza Strip homes next month, several weeks before the army is officially set to withdraw from the coastal area.
The 40 settler families from northern Gaza make up only a
small fraction of the 1,600 families slated for evacuation
But Saturday's announcement is the latest sign that Jewish settlers are coming to terms with Prime Minister Ariel Sharon's decision to evacuate the Gaza Strip and four West Bank settlements.
At least 500 families have already reached agreement with
the government on compensation and relocation, but none
had officially announced when they would move.
Many other Jewish settlers have vowed to resist the evacuation to the bitter-end.
In the West Bank town of Hebron, Israeli soldiers have evacuated three Palestinian schools that they had used for four years as posts to observe resistance fighters.
The army said it left the buildings to "ease restrictions on residents" and after reassessing the security situation.
Troops set up an observation post 200m from one of the schools, but said Palestinians will be allowed to reach and enter all of them freely.
Palestinian security forces are
trying to quell rising violence
Palestinians walked across broken glass from smashed windows and pushed aside shattered doors as they assessed the damage to the schools, once frequented by more than 2000 students.
Officials plan to rebuild and open them in a month.
Troops seized the schools, in a Palestinian-controlled area overlooking a Jewish settlement, after a Palestinian uprising erupted in 2000.
Israeli and Palestinian leaders declared a truce in February, although violence has flared in recent weeks.
Israel and the Palestinian Authority control different parts of Hebron, where about 500 heavily guarded Jewish settlers live among 120,000 Palestinians. It has often been the scene of deadly clashes.
"My four daughters went to the school and I was worried about my four daughters walking among soldiers and settlers to reach other schools," said resident Mohammed al-Jabari.
"I am glad to hear the students will be able to resume their normal studies," said governor Arif al-Jabari. "But Hebron is in a miserable situation. The occupation still exists."
Palestinians say Israel restricts their freedom of movement in the city with closures and roadblocks.