According to the report, Hamas politburo chief Khaled Mashaal was ordered by Assad to release kidnapped IDF soldier Gilad Shalit.
The report, which so far does not have confirmation in other sources, says that Mashaal and other Hamas top figures rejected Assad's demand, and left Damascus for Algeria.
PNN says conversation between Assad and Mashaal was held after the Air Force sent warplanes to carry out a flyby over the palace of the Syrian president in Latakia.
According to the internet site, Assad asked Mashaal to offer a solution to the crisis, to release the soldier, and to respond positively to the Egyptian mediation efforts, but the head of Hamas in Syria rejected the request and left the Syrian capital for Algeria together which his family and deputy Musa Abu Marzouk, and Marzouk's family.
Most of Hamas' politburo members also left Damascus.
The al-Arabiya TV news network later said that a senior Syrian source confirmed that Hamas politburo head Khaled Mashaal and other senior Hamas members have left Syria.
According to the report, the senior source in Syria reported that Mashaal left the country three days ago. The station cited Hamas sources denying the report, however.
Contacts on Shalit continue
A few days ago, the Lebanese Al-Nahar newspaper reported of estimates that Mashaal left Syria for an Arab country in North Africa, the name of which was not mentioned. The paper said that senior organization members were taking increased precautions for fear that they will be targeted.
According to the report, Mashaal and his people left Damascus for fear they will be hurt, but the PNN report shed new light on the Hamas members' departure from Damascus and on their relations with the Syrian president.
Hamas' spokesman in the Gaza Strip, Mushir al-Masri, responded in a conversation with Ynet to the report, saying that he was not familiar with the incident and cannot address it. He added, however, that he believed the information was not true.
Senior Palestinian sources in the Palestinian government say that contacts and mediation efforts over the kidnapped soldier continue.
Government spokesman Dr. Razi Hamad said that the Egyptian security delegation is continuing to offer solutions and ideas in order to bring about a breakthrough between the two side's entrenched stances.
Hamad said that one international source is involved in mediation efforts, and that the central idea being discussed is the release of the kidnapped soldier in exchange for the release of Palestinian prisoners at a later date.
Palestinian government Secretary General Muhammad Awad said that contacts were ongoing.
He added that the international community must pressure Israel to end its attacks on Gaza in order to create an atmosphere appropriate for continued discussions. Describing the IDF's activity in Gaza as the Israeli escalation, Awad said it prevented an obstacle to more contacts to free the kidnapped soldier. The Palestinian government called on the kidnappers to keep Gilad Shalit alive.
Roee Nahmias contributed to the report