Abbas, who leads the opposition Fatah party, was due to meet Meshaal in an attempt to end growing conflict between the two rival groups.
The meeting was to take place after Abbas had talks with Bashar al-Assad, the Syrian president.
Abbas and Meshaal were expected to discuss the names of the ministers for interior, finance and foreign affairs in a proposed Palestinian unity government.
Fatah and Hamas have tried for months to agree on a national unity government, in hope of ending a boycott by the United States and the European Union and in order to present a united front towards Israel.
Washington and Brussels consider Hamas to be a terrorist group and suspended direct financial aid to the Palestinians because the movement refuses to renounce violence or recognise Israel.
Tensions between the two factions, which had already claimed a number of lives, boiled over in the Gaza Strip in December, killing more than 30 people in that month alone.
However, violence has tapered off in the past couple of weeks.
Clashes last month began after Abbas called for early elections as a way of resolving the standoff with Hamas, which rejected the move.
Meeting with Syrian leader
Abbas's meeting with al-Assad reportedly ended amicably with the Syrian leader reaffirming his commitment to the Palestinian cause.
Briefing reporters in Damascus, Nabil Amir, Abbas's spokesman, said the two leaders had an "amicable and very frank" discussion that focused on "Palestinian issues in detail and the regional situation".
He reported that al-Assad had told Abbas that he was "attached to Palestinian national unity and ready to support anything that could help reinforce it, as well as efforts to find an adequate solution to the Palestinian problem."
Syria hosts the exiled leadership of a number of Palestinian groups and could exert considerable influence over Hamas. Meshaal moved to Syria from Jordan after an Israeli assassination attempt in 1997.
Source: Al Jazeera and agencies