Last Updated 9:12 a.m. ET
(CBS News) DAMASCUS, Syria - U.N.-Arab League envoy Kofi Annan held a second round of talks with Syrian President Bashar al-Assad Sunday in hopes of securing an encouraging response to his high-profile mission: To arrange a national political dialogue between the government and the opposition and gain unfettered access for humanitarian aid agencies.
Annan appeared to make little headway in his first round of talks with Assad on Saturday, when the Syrian president said rejected any political dialogue as long as "terrorist" groups would continue to try to destabilize the country.
According to a U.N. official in Damascus who spoke to CBS News on condition of anonymity, Annan ended his talks Sunday with strong hopes he managed to make an advance in solving Syria's year-long crisis. "He ended very candid, positive talks with Assad, and he feels very optimistic that he is making a breakthrough," the U.N. source told CBS News.
The official said Annan will head to Qatar, Cairo and New York before returning "pretty soon" to Damascus.
Meanwhile, opposition groups have dismissed the offer for dialogue while the Syrian military continues its offensive in the north. Burhan Ghalioun, head of the Syrian National Council, the most prominent opposition group in exile, said Friday that calls for dialogue were "naive."
Syrian forces continued to shell opposition strongholds in Homs, while fierce fighting was reported in the north Syrian town of Idlib on Saturday, where fighters from the Free Syrian Army were trying to hold back government troops.
Sixteen rebel fighters, seven soldiers and four civilians were killed in the Idlib fighting, according to the Syrian Observatory for Human Rights, which said 15 other people, including three soldiers, had been killed in violence elsewhere.
U.N. sources said Annan restarted his talks with Assad after having met with Syrian religious leaders, including the grand mufti Sheikh Bader El-Deen Hassoun, the senior Sunni Muslim authority, and the Greek Orthodox Christian patriarch Zaka Iwas.
The visit to Damascus by Annan, a former U.N. Secretary-General, is the centerpiece of a high-profile international attempt to find a solution to the worsening conflict amid sharp divisions among world powers and Arab countries over how to deal with the crisis.
On Saturday, Annan met leaders of some Damascus-based opposition groups, including Hassan Abdul Azim, General Coordinator of the National Coordination Committee for Democratic Change; Abdel Aziz al-Khair, a member of the National Coordinating Body for Democratic Change; Louay Hussein, head of the recently-formed opposition Trend for Building the Syrian State; and former opposition parliamentarian Riad Seif.
Following his 50-minute meeting with Annan, Azim told CBS News that that "any negotiation for a transitional stage demands first of all a ceasefire, end of violence, and the release of detainees."
The prominent opposition figure added that there "could be no negotiations under violence."
Azim said his delegation had asked Annan to coordinate with the Arab League. "We thanked him for his efforts and told him that the crisis is complicated and has already got into dangerous stages that threaten the unity of the country," he said.
Khair said he believes Annan is seeking to avoid any possible militarization or intervention into Syrian affairs. "The main challenge for Annan in this regard is to push the (Assad) regime to ceasefire, address the military aspects, and after that providing the relief needed by the Syrian people . . . and then comes the process of securing the necessary needs for launching a political process."
Annan has distanced himself from military intervention, as did the opposition members, agreeing that an armed conflict would only worsen the predicament of civilians.
The United Nations said more than 7,500 have died in the past year, and at least one activist group said more than 9,000 people have been killed.
The Syrian regime has insisted it is not killing dissidents but rather armed thugs who they say are responsible for killing thousands. It said in December that "terrorists" had killed more than 2,000 soldiers and police.