Saleh was operated on twice on Sunday, a Saudi official in Riyadh told AFP.
"President Saleh underwent two operations that were successful. The first was to remove a piece of shrapnel from his chest, and the second was neurosurgery to his neck.
"The next procedure will be for cosmetic surgical purposes. The period of convalescence is two weeks, after which he will return to Sanaa," the official said on condition of anonymity.
Saleh, 69, wounded by an explosion as he prayed at a mosque inside the presidential compound on Friday, was transferred to Saudi Arabia late on Saturday, but he has not stood down.
He was flown to Riyadh on a Saudi medical aircraft and taken immediately to a military hospital, while a second plane carried members of his family.
Saleh suffered "burns and scratches to the face and chest," a Yemeni official had said, playing down the extent of his injuries, after the GPC said he was "lightly wounded in the back of the head."
His regime has blamed the attack on powerful dissident tribal leader Sheikh Sadiq al-Ahmar, whose fighters have been battling government forces in Sanaa since a power transfer plan crumbled last month.
Sheikh Sadiq's office said on Sunday he has agreed to a conditional truce and to pull his forces from public buildings, following a request from Vice President Abdrabuh Mansur Hadi.
Hadi asked for "a ceasefire and the evacuation of public buildings by armed elements... which the sheikh accepted on condition that military units and their armed groups also pull back to restore security," an official in Sheikh Sadiq's office said.
Hadi had sent two generals with the truce offer, the official said.
Political police chief General Ghaleb Qamash and presidential military adviser General Mohammed Qassimi proposed "dismantling all military and recent positions created in the Al-Hassaba and Hada districts in the north and south of the capital respectively," the official added.
The veteran Saleh's eldest son Ahmed, commander of the elite Republican Guard, has remained in Yemen. The opposition says Ahmed was already preparing to take over from his father before a popular uprising broke out in late January.
Saleh's sons must be "forced to hand power over to Abdrabuh Mansur Hadi," who under the constitution replaces the absent Saleh, according to Qahtan.
"We are ready to cooperate with Abdrabuh but the problem is whether his (Saleh's) children are ready to hand power over to him," said Qahtan.
A source close to the presidency told AFP that the vice president and Ahmed Saleh held a top-level meeting with several military officials late on Saturday, without disclosing the outcome.
Hadi also met the US ambassador to Sanaa, Gerald Michael Feierstein, the state news agency Saba reported, to discuss "the importance of cooperation with the (opposition) Common Forum" alliance.
On the ground, tens of thousands took to Sanaa's streets to celebrate what they said was the end of Saleh's 33-year-long autocratic rule.
"Today, Yemen is newborn," sang people massed in University Square -- dubbed "Change Square" -- the epicentre of anti-regime protests that have raged since January.
"This is it, the regime has fallen," they chanted, as demonstrators slaughtered sheep for a traditional celebration.
In Yemen's second-largest city Taez, a flashpoint of demonstrations south of Sanaa, hundreds also took to the streets, shouting: "Freedom freedom, Ali has fled."
Saleh has refused to give up power despite four months of anti-regime protests that have left at least 200 people killed across the country.
His security forces have waged a brutal crackdown, while he has repeatedly refused to sign a plan brokered by Arab monarchies in the Gulf for a peaceful transfer of power.
Meanwhile, separate clashes between gunmen and security forces in Taez on Sunday left four soldiers and three gunmen dead, witnesses and a medic said.