Yet nor is it clear that even Mr Meshal has any control over events.
By contrast, Khaled Meshal, the movement's supreme leader in exile in Syria, says the opposite.
Messrs Abbas and Meshal have agreed on a new election commission to prepare for Palestinian elections in 2012.
ECONOMIST: Palestinian unity: Rivals who may need each other | The
With Hamas's centre of gravity shifting back to Gaza via Cairo, Mr Haniyeh may be eyeing Mr Meshal's job.
Yet Russia's president, Dmitri Medvedev, recently held talks with Mr Meshal, making it plain he intends to foster relations.
Egyptian efforts to negotiate the soldier's release, which some Hamas politicians openly supported, have been torpedoed by Mr Meshal.
King Hussein has said that after the attempt on Mr Meshal's life he had considered breaking off relations with Israel.
Sheik Meshal says he then turned to landing UBS a lead advisory role on a sale to a different buyer.
Mr Meshal also points to a split in the peacemaking Quartet of the EU, Russia, the UN and the United States.
One suggested that Mr Meshal's words had been distorted, while another hastened to point out that Hamas's position hadn't actually changed.
ECONOMIST: Israel exists. A Hamas leader briefly admits the obvious
When Sheik Meshal asked for his payment, UBS balked, his testimony says.
To keep Mr Abbas on the defensive, Hamas's leaders in Gaza have sought to block the joint programme he negotiated with Mr Meshal.
But in a recent interview, Khaled Meshal, the head of Hamas, implied that the organization's charter, calling for the destruction of Israel, is outdated.
The portly Mr Haniyeh is encroaching on Mr Meshal's turf, touring the region with a posse of ministers from Gaza, to be hosted by heads of state.
At the time of the assassination attempt on Mr Meshal, King Hussein said the future of his peace with Israel hung on whether the Hamas leader lived or died.
As for Mr Meshal, having realised to his chagrin that Gaza was no longer his to bargain away, he declared that he would retire after 15 years in the job.
Sheik Meshal then describes how he says he used his influence on UBS's behalf, placing a series of telephone calls that he says led to the demise of Vivendi's bid.
So Mr Meshal is waiting for the talks to fail, hoping for a groundswell of diplomatic calls for Hamas to be part of future negotiations, even if it refuses to recognise Israel first.
In his testimony, Sheik Meshal said that Omar Al Salehi, the UBS vice chairman of investment banking in the Middle East, offered him the fee at a meeting in mid-2009 when enlisting his help on the Zain deal.
Sheik Meshal says that before the first meeting even ended, he talked to another member of the Kuwaiti royal family who was then one of Zain's biggest shareholders, and got him to agree to help, according to the testimony.
It is difficult to know if these kinds of statements represent any kind of opportunity for negotiation, given the history of Meshal and Hamas, but sometimes a few words in the Middle East do indicate a potential for renewed talks.
Although Mr Meshal has more weight, plenty of other Hamas leaders have said many times that Israel is a reality, that they want a Palestinian state along the 1967 ceasefire lines (ie, in the West Bank and Gaza only), and that one day, perhaps after a few decades of peaceful coexistence, they might put the question of formally recognising Israel to a Palestinian referendum.
ECONOMIST: Israel exists. A Hamas leader briefly admits the obvious
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