Yemeni and Saudi officials say Mr Saleh will return to Yemen, but analysts say that is in doubt Thousands of people in Yemen have been celebrating his departure after weeks of anti-government protests.
It remains unclear whether Mr Saleh will return to Yemen.
There were reports he would remain in Saudi Arabia for two weeks; one week to recover and another for meetings, but it was not known what he planned to do after that.
Yemen's deputy information minister, Abdu al-Janadi, said Mr Saleh would be returning.
"Saleh is in good health, and he may give up the authority one day but it has to be in a constitutional way," he said.
Power struggle Mr Saleh underwent two successful operations on Sunday on his chest and neck, Saudi officials said, suggesting he would seek to return to Sanaa after convalescing.
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The CIA reckons al-Qaeda in the Arabian Peninsula (AQAP) is now as dangerous as the al-Qaeda grouping based in Pakistan.
AQAP has been responsible for a number of attacks, including the attempt to bring down an airliner heading to Detroit on Christmas Day in 2009 and to blow up planes using bombs hidden in printer cartridges the following year.
The radical cleric Anwar al-Awlaki is hiding somewhere in the country and he's been highly effective at radicalising individuals in the West using modern media, including the woman who stabbed British MP Stephen Timms.
President Saleh worked with Washington in countering al-Qaeda, although not always publicly, and within limits.
He always claimed that those who opposed him at home were less likely to co-operate with the West.
Washington also invested heavily in personal relationships but in recent months it has feared that the chaos engulfing the country was only strengthening al-Qaeda's hand.
Quite what is coming next in Yemen is unclear and what that means for dealing with al-Qaeda.
Analysis
There's growing concern that an al-Qaeda group based in Yemen has been using the political instability in the country to re-arm.The CIA reckons al-Qaeda in the Arabian Peninsula (AQAP) is now as dangerous as the al-Qaeda grouping based in Pakistan.
AQAP has been responsible for a number of attacks, including the attempt to bring down an airliner heading to Detroit on Christmas Day in 2009 and to blow up planes using bombs hidden in printer cartridges the following year.
The radical cleric Anwar al-Awlaki is hiding somewhere in the country and he's been highly effective at radicalising individuals in the West using modern media, including the woman who stabbed British MP Stephen Timms.
President Saleh worked with Washington in countering al-Qaeda, although not always publicly, and within limits.
He always claimed that those who opposed him at home were less likely to co-operate with the West.
Washington also invested heavily in personal relationships but in recent months it has feared that the chaos engulfing the country was only strengthening al-Qaeda's hand.
Quite what is coming next in Yemen is unclear and what that means for dealing with al-Qaeda.
But even if President Saleh wants to return, it is unlikely Saudi Arabia will allow him, BBC Middle East correspondent Jon Leyne says.
Yemeni Vice-President Abd-Rabbu Mansour Hadi has replaced Mr Saleh in his absence, and is in command of the armed forces and security services.He met US ambassador Gerald Michael Feierstein, state news agency Saba reported, to discuss "the importance of co-operation with the [opposition] Common Forum" alliance.
He may have little real power however, with Mr Saleh's son and other relatives in charge of key units of the security forces.
Friday's attack on Mr Saleh came after days of street battles in Sanaa between government forces and fighters loyal to Sheikh Sadiq al-Ahmar, head of the powerful Hashid tribal federation. The fighting has left more than 160 dead and brought the country to the brink of civil war.
Early on Monday a truce between the two sides appeared to be holding.
That power struggle overlaid widespread street protests that began earlier in the year, inspired by the uprisings in Tunisia and Egypt, calling for democratic reforms and an end to Mr Saleh's rule.
As word spread that Mr Saleh had left for Saudi Arabia, thousands of people in Yemen began celebrating.
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The BBC's Lina Sinjab says that it remains unclear if Mr Saleh will return, but anti-government protesters are already celebrating victory in Sanaa
President Saleh - who at times used brute force to try to quell demonstrations - had agreed to a deal brokered by the Gulf Co-operation Council that would see him step down in return for an amnesty from prosecution.
However, he has so far refused to sign the deal.
Some analysts say Saudi Arabia will now use his presence in the country to pressure him to sign the agreement.
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