Annan has flown to Iran for talks with the main regional ally of Damascus.
The former UN secretary general is trying to rescue his six-point peace plan, which was worked out with the Syrian government and rebels in April but faltered because the ceasefire it was supposed to begin with never took hold.
Major powers agreed at a meeting with Annan on June 30 that a transitional government should be set up in Syria, but remain at odds over what part Assad might play in the process.
"I just had a positive and constructive discussion with President Assad," Annan said before leaving for Tehran.
"We agreed an approach which I will share with the opposition," he told reporters in Damascus. He gave no details, but again stressed the importance of halting violence that has killed over 15 000 people in 16 months, by an opposition count.
Syrian foreign ministry spokesperson Jihad Makdissi said in a Twitter message: "In both meetings we reassured Annan of Syria's commitment to implement the six-point plan and hoped [the] other side is mutually committed."
In a TV interview on Sunday, Assad said he remained committed to Annan's plan and accused the United States, Saudi Arabia, Qatar and Turkey of supplying arms and logistical support to insurgents fighting to end 42 years of Assad family domination of the pivotal Arab state.
"We know that [Annan] is coming up against countless obstacles but his plan should not be allowed to fail, it is a very good plan," he told Germany's ARD network.
"The main obstacle [is] that many countries don't want [it] to succeed. So they offer political support and they still send armaments and send money to terrorists in Syria," Assad said, according to a transcript of the interview, held in English.
Syria, led by members of the Alawite sect, related to Shi'ite Islam, has alleged that the Sunni Muslim-led Gulf monarchies are backing unrest among its Sunni majority to check Shi'ite influence in the region, notably that of Shi'ite Iran.
Russia, which has so far defended Assad from the threat of UN sanctions, said it would not deliver Yak-130 fighter planes or other new arms to Syria while the situation there remained unresolved.
"While the situation in Syria is unstable, there will be no new deliveries of arms there," said Vyacheslav Dzirkaln, deputy director of the Federal Service for Military Technical Cooperation, told journalists at the Farnborough Air Show in Britain, according to the Interfax agency.
The refusal to send more arms to Syria – a trade dating back to the Soviet era – may signal the most pointed move yet by Moscow to distance itself from Assad as rebels have gained some ground and the death toll climbs.
Annan's plan calls for an end to fighting by government security forces and rebels, withdrawal of the government's heavy weapons from towns, return of the army to barracks, humanitarian access, and dialogue between the government and opposition aimed at a "political transition".
Opponents of Assad invited to Moscow for talks insisted that the political dialogue Annan is trying to initiate must start with a change at the top – a position that Russia rejects, and Annan's plan does not specify.
"The transition period must begin with Assad's departure," said Samir Aita, representative of the opposition Democratic Syria Foundation. "A national government must be created, and in order to do that, an all-Syrian national conference, where all representatives of the Syrian population would express their opinions, needs to take place."
Anti-Assad activists in Syria reported army shelling and clashes with rebels on Monday in Deir al-Zor, Deraa, Homs, Aleppo and a neighbourhood of Damascus. Residents reported the sound of gunfire in the capital. An activist website said over 100 Syrians had been killed on Sunday, most of them civilians.
US secretary of state Hillary Clinton said on Sunday that opposition forces were growing more effective, and the sooner the violence ended, the better the chances of sparing Syria's government a "catastrophic assault" by rebel fighters were.
While Assad has faced sanctions and international condemnation over his crackdown on dissent, major Western and Arab powers have shied away from direct military action.
British foreign secretary William Hague said in a speech in the Netherlands: "We will continue to try to persuade Russia and China, but if the Kofi Annan plan fails no option to protect lives will be off the table."
Assad told his German interviewer he did not fear that he might share the fate of the Libyan leader Muammar Gaddafi, who was killed after capture, or Hosni Mubarak, the toppled Egyptian president sentenced to life imprisonment.
Assad said that most of the victims of the uprising were supporters of the government.
"From the list that we have, from the names that we have, the highest percentage are people who are killed by gangs, different kinds of gangs … If you talk about the supporters of the government – the victims from the security and the army – are more than the civilians," he said.
Activists, who keep lists of names and dates of death, and Western governments say more than 15 000 people have been killed by forces loyal to the government, the great majority of them people who opposed the government and their innocent families.
Syria says more than 2 600 members of the security forces have been killed. – Reuters