Annan's talks in Damascus come a day after activists said Assad's forces killed at least 68 people as they sought to extend control over the rebellious city of Homs and crush armed opposition in the northern province of Idlib.
Arab foreign ministers were also due to hold talks in Cairo with Russian Foreign Minister Sergei Lavrov. Russia and China vetoed a UN draft resolution last month which would have backed an Arab League plan calling for Assad to step aside.
Activists said tank rounds and mortar bombs crashed into opposition districts in the rebellious central city of Homs, killing 20 people, while 24 were killed in the northern province of Idlib and more deaths were reported elsewhere.
"Thirty tanks entered my neighborhood at seven this morning and they are using their cannons to fire on houses," said Karam Abu Rabea, a resident in Homs's Karm al-Zeitoun neighborhood.
One focus of demonstrations was the anniversary of Kurdish unrest inSyria in 2004 when about 30 people were killed.
Many thousands of Kurds demonstrated in northeastern cities, YouTube footage showed, some carrying banners that read "Save the Syrian people". Other clips showed hundreds of protesters in the Assali district of Damascus, burning posters of Assad's father Hafez al-Assad and chanting "God damn your soul, Hafez".
Street protests have swelled every Friday after Muslim prayers since the anti-Assad revolt erupted a year ago, despite violent repression by the military and loyalist militias.
Decisive victory has eluded both sides in an increasingly bloody struggle that appears to be sliding into civil war.
"Immediate ceasefire"
The former UN chief Annan, now the UN-Arab League envoy, discussed his mission with current UN Secretary General Ban Ki-moon and the head of the Arab League, Nabil Elaraby.
"I have very strongly urged Kofi Annan to ensure that there must be an immediate ceasefire," Ban told reporters in New York after the conference call. After a ceasefire, he said, there should be "inclusive political solutions" found through dialogue.
Annan also plans to meet the Syrian opposition before leaving the country on Sunday. He has called for a political solution, but dissidents say there is no room for dialogue amid Assad's crackdown.
"If (Annan) can persuade Russia to back a transitional plan, the regime would be confronted with the choice of either agreeing to negotiate in good faith or facing near-total isolation through loss of a key ally," the Brussels-based International Crisis Group said in a paper this week.
Along with China, Russia has opposed any UN resolution, fearing Libya-style military intervention, a position Germany hoped might change after Vladimir Putin's victory in the presidential election on Sunday.
"I hope that Russia has a clearer view after the election. It is crucial that the UN Security Council gives a clear statement that shows that we stand by the people of Syria and are against violence and repression," German Foreign Minister Guido Westerwelle told reporters at an EU meeting in Denmark.
Russia, an old ally of Damascus and its main arms supplier, has defended Assad against his Western and Arab critics, twice joining China in vetoing UN resolutions on Syria.
A Russian diplomat said Assad was battling al Qaeda-backed "terrorists" including at least 15,000 foreign fighters who would seize cities if government troops withdrew.
The Syrian opposition denies any al Qaeda role in the uprising, but Assad's opponents have taken up arms. The British-based Syrian Observatory for Human Rights said five soldiers were killed when their armored personnel carrier was attacked by army deserters in the southern province of Deraa.
No equivalence
France's Foreign Ministry said Paris would not accept any UN Security Council resolution which would assign responsibility for the violence in Syria equally between the Syrian government and the opposition.
"There is no equivalence between the savage repression that Bashar al-Assad's clan has perpetuated for months and the legitimate desire of the Syrian people for the respect of their rights," said ministry spokesman Bernard Valero.
Foreign Minister Alain Juppe denied Russian suggestions the West was seeking a pretext for military action: "The option of any military intervention is not on the table," he said.
China, which despatched an envoy to Syria this week, said on Friday it would send an assistant foreign minister to the Middle East and to France to discuss a way forward.
Beijing urged Annan to "push for all sides in Syria to end their violence and start the process of peace talks".
UN humanitarian chief Valerie Amos, who visited Homs this week, said Assad's government had agreed to join UN agencies in a "limited assessment" of civilian needs in Syria, but had not met her request for unhindered access for aid groups.
Syrian officials had asked for more time, she told a news conference in Ankara after visiting Syrian refugees arriving in growing numbers in border camps in Turkey.
Amos said she was "devastated" at the destruction she saw in Homs and that she wanted to know the fate of civilians who had lived in the city's Baba Amr district, which rebel fighters left on March 1 after a 26-day siege.
Syrian security forces have killed well over 7,500 people since the anti-Assad uprising began a year ago, according to a UN estimate. The government said in December that "armed terrorists" had killed more than 2,000 soldiers and police.
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Kofi Annan seems to be on a very difficult mission. He had won a Nobel Peace Prize in 2001 for - mostly - restoring public confidence in the U.N. organisation. This mission will be a litmus test for his diplomatic skills. Could he come up with a peaceful solution to the crisis, he might deserve another Nobel Prize.
Syrian security forces have killed well over 7,500 people since the anti-Assad uprising began a year ago, . Shame on anyone who is protecting Assad. When a Dictator decides to use artillery on his own cities it's time for him to go.