The meeting hosted by Qatar's emir was a "follow up" to Thursday's talks, which lasted nearly three hours, the source said on condition of anonymity.
The discussions, at the palace of Qatari ruler Sheikh Tamim bin Hamad al Thani, came after fighting in Gaza flared anew on Tuesday as Egyptian-brokered truce efforts collapsed.
They broke down with Israel insistent on its demand for security from rocket fire by Gaza militants, and Hamas defiant over its call for an end to eight years of Israeli blockade.
In Qatar, no information was immediately available on the content of the discussions there. The Palestinian source said Abbas would leave Doha after the meeting.
The Hamas armed wing declared the truce efforts over Wednesday after Israel carried out an abortive assassination attempt on its leader Mohammed Deif, killing his wife and two of his children.
Four Palestinians were killed in new Israeli air strikes on Gaza on Friday, raising the death toll since July 8 to 2,087 Palestinians dead and 67 Israelis.
The war is the deadliest confrontation between Israel and Hamas since the second Palestinian intifada, or uprising, between 2000 and 2005.
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@Hyperbaig: Hmmm, And just where might all of us find that "all evidence" you speak of! Curious that you somehow neglected to include that source!
@someone: maybe because all evidence suggests the killing had nothing to do with Hamas. I would suggest you stop defending the indefensible terrorist state of Israel.
I wonder why Pakistani news paper did not publish news about Hamas's military wing Al Qassim claiming killing of those three Israeli teenagers which caused all this violence.