Al Jazeera reported that armed men attacked the Gaza Islamic University, a Hamas stronghold, late on Thursday night.
Hamas said its fighters ambushed what it said was a convoy carrying weapons to Abbas's Presidential Guard in the Gaza Strip.
A Fatah spokesperson said the convoy was carrying items for civilian use only.
Al Jazeera's correspondent Nour Odeh said gun battles were continuing and spreading to northern Gaza.
"We are hearing very loud explosions," she said, adding that Fatah supporters were raiding the university.
In northern Gaza, a security compound belonging to the Presidential Guard was under siege by armed Hamas men, she said.
Residents said it was the fiercest fighting since the truce went into effect on Tuesday, highlighted by the burning down of a Palestinian Authority military compound by Hamas fighters.
Agencies reported quoting a security source that bodyguards of Abbas occupied the Gaza Islamic University, adding that several armed men were arrested and weapons and explosives confiscated after a search of the campus.
The Presidential Guard decided to move in after Hamas fighters there fired mortar shells and rocket propelled grenades at the offices of the Palestinian presidency in Gaza City.
Laila El-Haddad interviews members of armed groups allied to Hamas and to Fatah to discover why they do what they do - read An eye for an eye in Gaza
Earlier on Thursday, heavy clashes took place outside the main entrance of the Bureij and Nuseirat refugee camps in the central Gaza Strip, security sources and witnesses said.
Nour Odeh said the situation deteriorated quickly and that people had closed their shops and gone home.
Hamas said that an "Arab country" had shipped weapons to the presidential guard which is loyal to the rival Fatah faction.
Ismail Radwan, a Hamas spokesman, said: "A large cargo of weapons was transported yesterday."
He said the shipment passed into the Gaza Strip via Egypt through a crossing on the border between Egypt, Israel and the Palestinian territory.
"These weapons are intended for the Presidential Guard and the Fatah putschists," Radwan charged.
"Military Jeeps and shells are among these weapons which come from an Arab country, whose name we do not wish to divulge."
Fatah denied having received any such shipment.
Abdelhakim Awad, a Fatah spokesman, said: "It's a slanderous accusation throwing water into the fire."
"It is pitiful to witness the bloody scenes in Gaza between the brothers"
Burhan Hammad, the head of Egypt's Gaza-based security delegation, also similarly branded the allegations a "total lie" that he said intended to push bitter rivals Fatah and Hamas into an "arms race".
The two factions have been locked in a power struggle since Hamas formed a government last March after a landslide victory over Fatah at the polls.