"There will be no meeting today," the Palestinian faction official told The Associated Press, speaking on condition of anonymity and that his faction not be named because of the sensitivity of the matter.
"There are difficulties facing such a meeting and there are current mediations to try and resolve the problems," he added, without elaborating. He said it was possible that the meeting would be held Sunday.
Abbas had been expected to hold crucial talks with Mashaal Saturday on forming a national unity government and ending months of Palestinian infighting that has claimed at least 62 lives.
There were hopes that the meeting — the first between Abbas and Hamas' exiled political chief since July 2005 — could end the year-long political deadlock between the militant Hamas and the more moderate Fatah.
But disputes still remain on the toughest issues — control of the two factions' powerful security forces and Hamas' refusal to recognize Israel's right to exist.
Before Abbas began meeting leading members of other Palestinian factions Saturday evening his aide Saeb Erekat appeared to downplay the significance of such a meeting and would not confirm it would happen.
"The purpose of this visit is for President Abbas to meet with President (Bashar) Assad, though I'm not excluding the possibility of President Abbas meeting with Mr. Mashaal and other Palestinian leaders in Damascus," he told reporters.
Abbas had billed the talks as a last-ditch effort to end the crisis among the Palestinians, threatening to call early elections if the two sides did not agree on forming a national unity government.
Abbas' Fatah Party and the Islamic militant Hamas have been at loggerheads since Hamas won parliamentary elections last year and formed a Cabinet.
Israel and Western countries imposed an economic boycott on the Palestinian Authority, demanding that Hamas renounce violence and recognize Israel's right to exist. The Islamic militant group has refused.
Abbas has been pushing Hamas for months to form a coalition government of independent experts, in hopes of ending the sanctions which have pushed Palestinians deeper into poverty.
With the two sides unable to agree, tensions have repeatedly exploded into open warfare in the Gaza Strip, a traditional Hamas stronghold.
Abbas came to Damascus bolstered by Israel's release to him Friday of $100 million which Israel had collected as taxes on behalf of the Palestinian government.
Following Hamas' election victory, Israel froze the funds. It later said the money would be routed to Abbas instead of the Hamas-controlled finance ministry, for humanitarian purposes and to strengthen Abbas' security forces which are on the front lines against Hamas.
Abbas was greeted at the Damascus airport by Foreign Minister Walid Moallem and then went to the presidential palace for talks with Syrian President Bashar Assad.
Palestinian legislator, Nabil Amr, who attended the meeting with Assad, later told reporters that the Syrian leader expressed readiness to support all "efforts aimed at achieving Palestinian unity."
Syria's official news agency SANA said Assad stressed the "necessity for the Palestinian people at this critical stage to reject internal fighting and strengthen their national unity." He said he supported all what the Palestinian factions agreed on and was ready to help in this regard, it added.
Syria hosts the exiled leadership of a number of Palestinian militant groups and could exert considerable influence over Hamas. Mashaal relocated to Syria from Jordan after an Israeli assassination attempt in 1997.
Abbas and Mashaal's frosty relationship worsened in April after Mashaal publicly accused Abbas of being a traitor for opposing the formation of a heavily armed Hamas militia.
For months, Abbas shunned all contact with the Hamas leader, and only recently agreed to take a phone call from him. Their meeting in Damascus could signal that a settlement was at near, Hamas officials have said.
Since last year's election, Hamas has controlled the Cabinet and parliament, while Abbas, elected separately, wields considerable power as president.
Abbas hopes a coalition Fatah-Hamas government would persuade the West to lift the painful sanctions imposed on the Hamas-led government for refusing to disarm and recognize Israel's right to exist.
Hamas has also said it would be willing to respect previous agreements between Israel and the Palestinian Authority that it deems fair to the Palestinians, something that falls short of full recognition of all previous accords as demanded by the international community and Israel.
That, and control of the Palestinian interior ministry which oversees the 85,000-strong security forces, are key obstacles to forming a coalition.