Syria, telling its members that Syria had agreed to the date, diplomats said on Monday.
The U.N.-Arab League peace envoy told the council behind closed doors that troops would stop entering Syrian towns and that there would be a withdrawal of heavy weapons and the start of a troop pullout.
A full ceasefire would be have to be in place by April 12 or 48 hours after the agreed date, Susan Rice, the U.S. ambassador to the United Nations, quoted him as telling the 15-nation council.
Annan met Syrian President Bashar al-Assad in Damascus on March 10 and presented him with a six-point plan calling for the military pullout. His spokesman said one week ago that Assad had accepted the terms, adding that the "the deadline is now".
One diplomat said Annan confirmed to council members that there had been "no progress on the ground" towards halting the violence, which continues with daily reports of army shelling and shooting, and clashes with the rebel Free Syrian Army.
"Today doesn't feel much different than yesterday or the day before, or the day before that," opposition activist Waleed Fares said from inside Homs. "Shelling and killing."
The Syrian Observatory for Human Rights, a British-based activist operation which collates reports from around Syria, reported 70 people killed on Sunday, including 12 civilian victims of shelling and sniper fire in Homs. 35 people were killed on Monday, SOHR said, including eight soldiers and nine rebels. Ten civilians were killed on Monday in the central province of Homs. In Syria's second city of Aleppo, a bomb blast at a kiosk killed the owner, an Assad supporter, it said. At least five people were killed and eight wounded in army bombardments of villages in northern Idlib province, which borders Turkey.
Turkish officials said refugees were crossing the border at a rate of around 400 a day. Over 40,000 Syrians have taken refuge in neighboring countries since the unrest broke out a year ago, according to U.N. figures.
Despite the lack of progress, Annan urged council members to "begin consideration of deployment of an observer mission with a broad and flexible mandate," a diplomat said.
The U.N. peace keeping department is already planning for a ceasefire monitoring mission that would have 200 to 250 unarmed observers. It would require a Security Council resolution.
It was not clear how Russia will respond to Annan's report. Russia and China have vetoed two council resolutions condemning Assad for turning the army on civilians demanding change.
NATO Secretary-General Anders Fogh Rasmussen repeated that the Western allies have "no intention whatsoever to intervene in Syria". He said he did not believe providing weapons would help.
Saudi Arabia and Qatar favor providing arms to the FSA. But most Arab states and Western backers of the rebels oppose that.
FSA rebels have said they will stop shooting if the army pulls heavy weaponry out of cities. But the Assad government has said it must maintain security in urban areas and there has been no sign of tanks, armor or artillery moving out.
The United Nations says Syrian soldiers and security forces have killed more than 9,000 people over the past 12 months. Damascus says rebels have killed 3,000 troops and police.
Assad blames the unrest on foreign-backed "terrorists" and has put forward his own reform program, which his domestic foes and international opponents have dismissed.
U.S. Secretary of State Hillary Clinton, speaking at a Friends of Syria conference with Assad's opponents in Istanbul on Sunday, said Assad had a long list of broken promises behind him and would face serious consequences if he did not halt actions targeting civilians.
Although Western powers have been wary of military intervention, Turkish Foreign Minister Ahmet Davutoglu compared the situation to Bosnia in the 1990s.
"In the case of Bosnia, the international community was too slow therefore we lost many people," he said. "In the case of Syria we have to act without delay."