Saleh said he would cede power only into "safe hands" and Yemeni political sources said talks were under way to work out the details of a peaceful transition.
But in Syria, protests challenging the rule of President Bashar al-Assad spread across the country after security forces killed dozens of demonstrators in the south.
"The barrier of fear is broken. This is a first step on the road to toppling the regime," said Ibrahim, a middle-aged lawyer in the southern Syrian city of Deraa. "We have reached the point of no return."
Saleh's departure would present a new challenge to Western countries already embroiled in a week-old military intervention in Libya, amid fears that instability in Saudi Arabian neighbor Yemen could open the way for al Qaeda to expand its power there.
"We don't want power, but we need to hand power over to safe hands, not to sick, resentful or corrupt hands," said Saleh, who had come under intense pressure to quit since snipers fired on anti-government protesters a week ago, killing 52 people.
That bloodshed prompted a string of defections that severely weakened Saleh's position, including by military figures such as top general Ali Mohsen, as well as diplomats and tribal leaders.
A source close to Mohsen said he and Saleh had discussed a deal in which both men and their families would leave Yemen, while political sources said broader talks were underway on a political transition.
A diplomat in the capital Sanaa, however, said it was premature to discuss an outcome. "It can go either way."
In Syria, Assad's government had promised on Thursday to look at giving greater freedom to Syrians.
But there was more bloodshed after Friday prayers, with witnesses reporting at least 23 dead, including three in the capital Damascus. Information on casualties was limited and authorities restricted journalists' movements.
HAULING DOWN ASSAD'S STATUE
In Deraa, tens of thousands marched in funerals for some of those killed earlier in the week, chanting "Freedom."
In a central square, a Reuters correspondent saw protesters haul down a statue of Assad's father, late president Hafez al-Assad, before security men in plain clothes opened fire with automatic rifles from buildings.
The crowd of some 3,000 scattered under volleys of bullets and tear gas. The reporter saw some wounded helped into cars and ambulances. It was unclear how many, if any, were killed.
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