Shara, 73, is the most powerful Sunni Muslim figure in the minority Alawite-led regime of President Bashar al-Assad and has served in top posts for almost 30 years.
Adding to the mystery, a former deputy oil minister who defected in March said Shara was actually under house arrest and that other top officials were being kept under surveillance.
"He has been trying to leave Syria," Abdo Hussameddin told Al-Arabiya television. "But there are a series of circumstances that prevent him from leaving, especially the fact that he has been under house arrest for some time."
Assad's regime has already been hard hit by a series of defections since the anti-regime revolt erupted in March 2011, including former prime minister Riad Hijab and high profile general Manaf Tlass -- a childhood friend of Assad.
"Initial reports show that there was an attempted defection, but that it failed," the rebel Free Syrian Army (FSA) said.
French Foreign Minister Laurent Fabius had said this week that there could be more "spectacular" defections from the regime, which was also rocked by a July bomb attack claimed by the FSA which killed four security chiefs.
The conflicting reports came as UN chief observer General Babacar Gaye accused both sides in the 17-month conflict of failing to protect civilians who have borne the brunt of the increasingly brutal violence.
-- 'Obligations to civilians have not been respected'--
"Both parties have obligations under international humanitarian law to make sure that civilians are protected," Gaye told reporters in Damascus ahead of the mission's end on Sunday. "These obligations have not been respected."
On the ground, as Muslims the world over geared up for Eid, the celebrations marking the end of the holy month of Ramadan, Syrians faced another daily cycle of bloodshed.
The military launched new air strikes on Aazaz in the northern province of Aleppo, three days after about 40 people were killed in the rebel-held town, according to the Syrian Observatory for Human Rights.
It also pounded areas of Aleppo as fighting raged between rebels and troops in the northern commercial hub which has become the focus of the conflict, partly because of its strategic location near the Turkish border.
In Damascus, fighting broke out in the heavily populated southern district of Tadamun, showing the rebels are still resisting despite government forces last month claiming they had retaken the capital.
Rebels also targeted a military convoy besieging the neighbouring Al-Hajar Al-Aswad district near the country's biggest Palestinian refugee camp, killing at least four soldiers and one rebel, the Observatory said.
It reported at least 99 deaths Saturday, including 12 rebels and five civilians killed in army shelling and fighting in Herak in the southern province of Daraa, cradle of the uprising.
And in a gruesome sign of the escalating brutality, the Observatory said dozens of bodies had been found dumped in several areas of Damascus province.
Accounts of people being shot dead by snipers or being executed are increasing, while government forces also appear to be resorting to more attacks from the air against the poorly armed and disparate rebel groups.
Overall the death toll has surged to at least 23,000 people, while the UN puts the toll at 17,000.
But the regime's far superior force has failed to suppress the rebellion -- whose fighters' determination to bring Assad down has only grown with the passing of time.
The intensified fighting has sent thousands more Syrians fleeing into neighbouring countries, particularly Turkey. The UN says that at least 170,000 have fled and another 2.5 million inside Syria need aid.
"This Eid, I don't want to celebrate," said Omar Shakir, a 21-year-old who escaped from the central city of Homs to Lebanon during the regime's relentless month-long onslaught in February on the then rebel-held district of Baba Amr.
"This Eid only serves a reminder for us that we are far from our families. There is nothing festive about it."
In a sign of a renewed effort to try to end the conflict, the United Nations announced on Friday the appointment of Brahimi as new Syria envoy.
But asked whether he was confident the civil war could be ended, Brahimi told France 24: "No, I'm not. What I am confident of is that I am going to try my utmost, my very, very best."
Brahimi's appointment won the backing of the United States as well as China and Russia, which have both vetoed Security Council resolutions on Syria and accused the West of hampering efforts to end the crisis.
Iran, Syria's closest ally, described the conflict as a struggle between Tehran and its arch-foe Washington, a statement that underscores Ban's stated fears it was becoming a "proxy war" between rival regional and international powers.