Fierce fighting continues to rage in the Syrian capital, Damascus, and in Aleppo, the country's commercial hub which has been battered by warfare between troops and rebels for the past three months.
Clashes erupted on Monday morning when troops tried to storm the rebel-controlled Damascus suburb of Harasta, activists said.
In Aleppo, violence reportedly flared in the southwest rebel district of Salaheddin, Izaa in the north and the central Old City.
The London-based Syrian Observatory for Human Rights said that on Sunday alone, 173 people were killed nationwide.
Joint UN-Arab League envoy Lakhdar Brahimi has been visiting Syria in a bid to establish a ceasefire during this week's four-day Eid al-Adha holiday, which begins on Friday.
Brahimi said he had contacted political opposition leaders inside and outside Syria and armed groups in the country. "We found them to be very favourable" to the idea of a truce, he said after meeting al-Assad on Sunday.
"We will return to Syria after Eid and if calm really takes hold during the feast, we will continue to work," he added.
Al-Assad for his part told Brahimi he was "open to any sincere efforts seeking to find a political solution to the crisis based on respecting Syria's sovereignty and rejecting any foreign interference," according to state media.
But hopes of a truce are "slim", a senior Arab League official said on Monday.
"The signs, both on the ground and by the government ... do not point to the presence of any real will" to implement a ceasefire, Arab League deputy secretary-general Ahmed Ben Helli told the AFP news agency on the sidelines of the World Energy Forum in Dubai.
"We are days away from Eid. We hope the situation changes and the government and opposition respond even a little bit to this door for negotiations." Ben Helli told Reuters news agency.
Russian Foreign Minister Sergei Lavrov urged the Syrian government and all opposition groups to accept Brahimi's proposal, calling it "a necessary step towards a long-term ceasefire and the launch of a political process aimed at providing for a Syrian democratic renewal".
Like al-Assad, Russia has laid most of the blame for the continuing violence on the rebels, who it says are aided by encouragement and weapons from abroad.
Russian deputy foreign minister Mikhail Bogdanov arrived in Tehran on Monday to discuss the crisis in mutual ally Syria with Iranian officials, Fars news agency reported. Iranian officials have welcomed Brahimi's ceasefire initiative.
The opposition says conflict, which has drawn in regional and international powers backing different sides, has killed more than 30,000 people since anti-regime protests erupted in March last year.
A Jordanian soldier was killed in clashes with fighters trying to cross the country's northern border into Syria, a cabinet minister said on Monday.
"The soldier died during an infiltration attempt," Minister of Information Samih Maaytah told Reuters, adding that it was the first death in Jordanian army ranks on the border since the uprising against Syrian President Bashar al-Assad broke out in March last year.
Troops detained 12 fighters after the clashes, which occurred overnight, an army statement said.
Jordanian authorities have privately expressed concern about the growing threat of jihadists infiltrating into Syria to join scores of radical groups operating across the country.
Jordan said on Sunday it foiled a plot by an al-Qaeda-linked cell to bomb shopping centres and assassinate Western diplomats.
Security sources said the 11 suspects detained, all Jordanians, had envisaged carrying out attacks in the capital Amman using smuggled weapons and explosives from Syria.
Security sources said the suspects had spent some time in Syria, without saying when they had returned to Jordan.
A key US ally in the Middle East and Israel's peace partner, Jordan enjoys close ties with Western intelligence
agencies and has often been targeted by al-Qaeda and other Islamist militants.