On Wednesday, Saleh welcomed Gulf mediation, according to state news agency Saba, which said he "affirmed the necessity of a serious and fruitful dialogue to overcome the current crisis."
The exit plan offered to Saleh would see him hand power over to his deputy, while providing guarantees of protection to him and his family, the opposition says.
A diplomat in Sanaa confirmed the content of the proposal, adding that it includes forming a national unity government led by the opposition.
The offer resembles one the opposition itself made at the weekend calling for Saleh to make way for Vice President Abdrabuh Mansur Hadi to head a caretaker regime.
A Yemeni opposition leader told AFP on Thursday that any efforts to help usher Saleh out of office were "naturally welcome."
"The ball is in Saleh's court and any developments resulting from any delay would be his responsibility," said Mohammed al-Sabri, a leading figure in the opposition Common Forum.
Tension was high in Sanaa on Thursday, with army units supporting the protesters camped near Sanaa University, while the Republican Guard, which has remained loyal to Saleh, was on high alert.
Streets were closed to traffic and checkpoints had multiplied in Sanaa, which is almost divided in half by rival forces, ahead of another day of demonstrations the government has dubbed a "day of reconciliation."
Saleh, a close US and Saudi ally in power since 1978, has faced nationwide protests since late January calling for his departure and around 125 people have been killed.
Despite Saleh being a key US partner in its fight against Al-Qaeda, Washington condemned the heavy-handed response by security forces to protests in the cities of Taez and Hudaydah, where around 20 demonstrators were shot dead this week.
"The United States strongly condemns the use of violence by Yemeni government forces against demonstrators in Sanaa, Taez and Hudaydah in the past several days," said White House spokesman Jay Carney.
GCC ambassadors have invited Saleh and the opposition to attend proposed talks in Riyadh, but a date has not been disclosed.
Both parties have welcomed the invitation, but the Qatari premier's statement appeared to show that Gulf states, which have been traditionally strong backers of Saleh, have decided to let him go.
In a statement on Saturday, the opposition stipulated the formation of a transitional national council, beginning a wide-reaching national dialogue and forming a committee of specialists to draft constitutional reforms.
It also demanded the establishment of a temporary national unity government led by the opposition and encompassing "all political parties, (demonstrating) youth and businessmen."
But the youth movement leading protests appeared unimpressed by proposed talks and handing power to Saleh's deputy, saying the president's departure has to come first.
"All the youth are against talks before the corrupt regime, headed by Saleh, falls," said Waleed al-Amary, a member of the media committee of the youth movement at a Sanaa protest.
On Thursday, thousands of demonstrators maintained sit-ins in Sanaa and in Taez -- Yemen's second city some 200 kilometres (125 miles) south.
Washington has expressed fears Al-Qaeda could take advantage of a prolonged crisis in Yemen, and has pressed Saleh to negotiate a transition of power.
"Yemen has really eased up the pressure on Al-Qaeda in the Arabian Peninsula," US Defence Secretary Robert Gates said in Iraq on Thursday, of the group's affiliate in Yemen.
But the Pentagon has said there are no plans to suspend US military aid to Yemen.
Amnesty International warned Wednesday against any immunity deal for those responsible for the deaths of dozens of anti-regime demonstrators in Yemen, especially the "Black Friday" bloodbath.
In that shooting on March 18, 52 people were gunned down in what the rights group called "an apparently coordinated sniper attack on a protest camp in Sanaa".
Yemen controls the Bab al-Mandab Strait, the strategic entrance to the Indian Ocean from the Red Sea, through which three million barrels of Gulf oil transits each day towards the Suez Canal and Europe.
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