Abbas's visit to Qatar was initially announced for Monday by Palestinian officials who are in Cairo for indirect talks with Israel on a lasting truce in Gaza.
Abbas will on Thursday discuss separately with Mashal and Sheikh Tamim bin Hamad al-Thani latest developments in the negotiations in Cairo and "aid and reconstruction" in Gaza, Palestinian ambassador Monir Ghannam told AFP.
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From Doha, Ghannam said Tuesday, Abbas will travel on to Cairo as part of contacts the Palestinian leadership is staging "with all the parties concerned" in the Israel-Palestinian conflict.
A new 24-hour ceasefire in Gaza will last until midnight on Tuesday (2100 GMT) under a last minute deal reached in Cairo on Monday night.
Egyptian mediators have been pushing both sides to put a decisive end to weeks of bloodshed in Gaza, which has killed more than 2,000 Palestinians and 67 on the Israeli side.
But few details have emerged of the state of the negotiations given the apparently irreconcilable demands of the two sides.
Palestinian sources claimed that the pending agreement includes seven sections (which have not been confirmed by Israeli sources):
1. Comprehensive end to blockade of Gaza, including all crossings, and international supervision over entry of construction materials
2. Rehabilitation of the Gaza Strip under the supervision of the Palestinian unity government
3. Provision of solution to electricity shortage within a year
4. Extension of permitted fishing zone from six to 12 miles within six months
5. Comprehensive end to economic blockade of the Gaza Strip
6. An agreement, in principle, regarding a seaport, but a postponement of discussion for a month from the signing of the accord; the proposal sets new discussions – in a month – on the technical and administrative operations of the future port
7. Postponement of deliberation on release of detainees for one month
However, the chief Palestinian delegate to truce talks with Israel warned on Tuesday that Gaza violence could erupt anew unless progress is made toward a lasting deal ahead of a midnight deadline in Egyptian-brokered talks.
The talks in Cairo center on an Egyptian proposal that meets some of the Palestinian demands, such as easing Israel's eight-year blockade on Gaza, but defer debate on other thorny issues until later.
The aim is to broker a long-term arrangement to halt more than a month of bloody fighting, although both sides have largely silenced their guns since August 4 thanks to a series of temporary truces.
"We must take advantage of every minute in the next 24 hours until we reach an agreement or the cycle of violence will continue," Azzam al-Ahmad, head of the Palestinian delegation, told reporters in Cairo.
The delegations were expected to resume talks at midday (0900 GMT), an official said.
Although the back-to-back truce agreements have brought relief to millions on both sides of the border, the drawn-out waiting and the fear of a resumption of fighting was beginning to test people's patience.
"No one here has any hope," said Riyad Abul Sultan, a father-of-10 with thick curly hair, smoking as he sits on a flimsy mattress at a UN school in Gaza.
"May be they'll finish the war for two hours, may be Israel will start bombing again."