The diplomatic face off came as five members of one family were killed in regime air strikes on a village in Damascus province, according to a watchdog.
The National Coalition, which many Western and Arab powers recognise as the sole representative of the Syrian people, was discussing the idea of a government-in-exile in Istanbul but differences have emerged over who should lead the new executive, an opposition official told AFP.
"A proposal was made to name Riad Hijab but it has run into much criticism," the official said on condition of anonymity.
Hijab, a former prime minister in the Assad regime, defected last August and has worked closely with Turkish leaders to help restructure the fragmented Syrian opposition.
The opposition bloc, which has called for the establishment of an interim government with full executive powers in rebel-held areas inside Syria, is due to meet on January 28 in Paris, French Foreign Minister Laurent Fabius said.
Syria's Foreign Minister Walid Muallem meanwhile struck a defiant tone, telling state television late on Saturday that those who demand the Assad removal want only bloodshed in the country.
"Nobody can afford to undermine the presidency -- it is unacceptable," Muallem said.
"The US continues to have the president's departure as a condition of regime change, ignoring the fact that the captain of a capsized ship does not jump into the first boat," he added.
But Assad's mother Anisa Makhluf has meanwhile left the war-torn country and joined her daughter in Dubai, Syrian expatriates in the United Arab Emirates and an activist said on Sunday.
Makhluf's "departure from Syria is another indication of Assad losing support even from within his family," said Ayman Abdel Nour, head of the newly formed group Syrian Christians for Democracy.
Analysts say Assad is increasingly relying on the tight-knit circle surrounding him, which includes Maher, his only brother still alive and who commands the army's notorious Fourth Brigade.
The opposition in Syria has insisted on Assad's departure as a prerequisite for any negotiations to settle the 22-month conflict that has killed more than 60,000 people, according to United Nations figures.
The UN Office for the Coordination of Humanitarian Affairs director John Ging has been in Syria since Friday to assess humanitarian needs in the war-ravaged country.
Ging aims to assess how much humanitarian help is needed in Syria, where an estimated four million people, half of them displaced, are in need of emergency assistance.
The OCHA mission was received on Sunday by Deputy Foreign Minister Faisal Muqdad and then travelled to Homs, even as a watchdog reported regime air raids on besieged districts of the embattled city.
In Damascus province, a couple and their three children were among nine civilians killed in air strikes on a village, while three people including a child died in shelling northeast of the capital, said the Syrian Observatory for Human Rights.
The Britain-based watchdog also reported one man killed by a sniper amid artillery shelling and warplane raids on Daraya, a strategic town next to Al-Mazzeh military airport east of Damascus.
Pro-regime daily Al-Watan said "terrorists on the outskirts of Daraya... have appealed for help after being hit very hard by the Syrian army, which destroyed several of the hideouts where they barricaded themselves."
It added the army would press on with a ground operation to "root out the remaining terrorists".