Al Jazeera’s Mohamed Vall, reporting from Damascus, said that the visit from the Palestinian president is significant because it highlights Syria's role as peacemaker in the Middle East.
Syria backs Hamas but has promised to help push for dialogue between the two sides.
"There are many factors that make this visit important... Syria is central in the Middle East, and it hosts a little less than three million Palestinian refugees here in Damascus ... many Palestinian factions ... and it hosts the political leader of Hamas, Khaled Meshaal. This makes Damascus central for the Palestinian issue," Vall said.
"Last Thursday, Khaled Meshaal met with president Assad and Hamas expressed its willingness to see Syria playing a role in creating reconcilition between Palestinian factions."
There were expectations that Abbas was going to meet Meshaal during this visit, however, according to Nabeel Abu Rudeina, the spokesman for the Palestinian presidency, Abbas' agenda does not include meetings with representatives from the Palestinian factions.
"There are no arrangements for any such meeting with Hamas leaders or others," Abu Rudeina told reporters in Jordan on Saturday. A Hamas official denounced the decision.
"The refusal of Abbas to meet with Hamas shows that he is complying with the US policies that rejects [inter-Palestinian] dialogue," the official told AFP.
"We are surprised by this position particularly since last month Abbas launched his initiative [for talks] and Hamas accepted it."
Vall reported that Syria has been stepping up its diplomacy in the region.
"Syria is trying to project a new image, showing that Syria is not only a troublemaker as it is seen in the West, but also a peacebroker and a peacemaker," he said.
"Syria simply wants to tell the West and the Arab world that it is trying to play a new role in the Middle East."