Mohamed al-Hindi, an Islamic Jihad leader, said he met Mohammed Dahlan, the powerful Fatah leader and member of the Palestinian Legislative Council, in Cairo.
Al-Hindi said: "I hope that these efforts will be rewarded by the signature of a national accord in Cairo in the coming days."
Separately, a source close to Abbas said he had received an invitation from Bashar al-Assad, the Syrian president, to visit Damascus and that Abbas was "not against a meeting with Meshaal".
Paving the way
The source said that Ziad Abu Amr, an independent Palestinian MP close to Hamas, and Mohammed Rashid, a former adviser to Yasser Arafat, the late Palestinian leader, had met with Meshaal to pave the way for a meeting.
Abbas and Meshaal have not met since Hamas formed a government in March 2006, after winning parliamentary elections.
Gaza has seen escalating violence since December 16, when Abbas called for early presidential and parliamentary elections, a move which Hamas rejected.
More than 30 people have been killed and hundreds have been wounded in clashes between the two groups.
Strike pressure

Bassam Zakarneh, the head of the government employees' union, said on Friday that workers would end a three-month long strike on Saturday after officials sign an agreement.
He told Reuters: "We have reached a deal to end the strike. The signing of the deal will be concluded tomorrow at President Abbas's office."
Palestinian officials said the agreement commits Abbas to start making partial salary payments immediately.
Last month, a similar deal fell through at the last moment.
Western diplomats said it is unclear where the money would come from.

The Palestinian authority's monthly wage bill tops $100 million. Workers have not received their full salaries since Hamas came to power in March 2006.