So if he moves ahead, it looks like Fatah could be having an election with itself.
Factional fighting between Fatah and Hamas claimed more than 90 Palestinian lives between December and February.
Abbas' Fatah group dominated Palestinian politics for decades until last year, when Hamas won legislative elections.
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The call comes after the reconciliation between Hamas and Fatah brokered last week by Egypt.
Fatah and the Palestinian Authority, led by Abbas, now govern only part of the West Bank.
His Fatah movement, based in the West Bank, is deeply split from Hamas, which governs Gaza.
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But the world will also watch closely how Mr Abbas treats the rival streams in Fatah.
Qaddoura Fares, one of Fatah's younger leaders in Ramallah, lets out a short, dry laugh.
Infighting between powerful members of Fatah has also held back reform of the security services.
For instance, they always say that the Fatah movement wants Hamas to recognize Israel.
It also took control of Fatah's intelligence files and the personal files of Fatah leaders.
But over the past week in Bethlehem, Fatah's leaders said they will not be appeased.
So do some moderates, such as Ahmad Ghunaim, one of the younger Fatah leaders.
Indeed, he said that Fatah agreed to add annexes to the agreement reflecting Hamas's positions.
They made the choice in March 2007 when Fatah and Hamas signed their first unity deal.
The statement ahead of the talks was criticised by a senior member of Fatah.
The Americans will tell the Israelis what cities and towns to transfer to Fatah control.
He did not state that the US is ending its training of the Fatah forces.
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The 90-day freeze is supposed to usher in a period of intense negotiations between Israel and Fatah.
Abbas' rival Fatah controls the West Bank and has been engaged in the peace process with Israel.
Mr Arafat, backed by most of his Fatah movement, insists the forces must remain answerable to himself.
In May 2011 Fatah and Hamas signed a reconciliation deal but it has not yet been implemented.
But this is no longer how younger Fatah activists in the West Bank and Gaza see things.
Mr Abbas does have majority support for the road map among his Fatah movement, say Palestinian sources.
The agreement between Fatah and Hamas makes Hamas a partner in the leadership of the Palestinian Authority.
If he does not do well, anger could spill out at a Fatah congress due in August.
Wexler based his claim on then-prime minister Ehud Olmert's 2008 peace offer to Fatah leader Mahmoud Abbas.
Some of the violence is factional fighting between Hamas and Fatah, or among Fatah's increasingly divided sub-factions.
As for Fatah, it is hard not to feel sorry for Fatah leader Mahmoud Abbas these days.
Secretary of State Condoleezza Rice arrived in Israel on Wednesday to advance the peace process with Fatah.
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