"The world is full of crazy people like me. So don't be surprised if Secretary General Ban Ki-moon can find someone who can do a better job than me," Annan said when asked if he thought someone else would be named to succeed him.
"There may be other plans, other approaches that may work quite effectively," he said, adding that at this stage the focus should still be on a political transition which means "President (Bashar) al-Assad will have to leave sooner or later".
Ban announced "with deep regret" the resignation of Annan, who was named to the post on February 23.
"Mr Annan has informed me, and the Secretary-General of the League of Arab States, Mr Nabil Elaraby, of his intention not to renew his mandate when it expires on 31 August 2012," Ban said in a statement, adding that he and Elaraby were in discussions on appointing a successor to Annan.
"Kofi Annan deserves our profound admiration for the selfless way in which he has put his formidable skills and prestige to this most difficult and potentially thankless of assignments," Ban said.
The statement added that Annan had said he would not renew his mandate when it expires on August 31.
The Security Council has repeatedly failed to reach a consensus on how to deal with the Syrian crisis, with Russia and China three times vetoing resolutions proposed by the West.
The former UN Secretary-General did succeed in getting Security Council backing for his six-point peace plan, but it failed to stop Syria's slide into civil war.
The plan, which called for a daily ceasefire and immediate negotiations between the regime and opposition groups, was not implemented and both sides ignored an April 12 ceasefire.
The Syrian government said Annan's decision was regrettable and accused "countries which seek to destabilize Syria" of having "hindered and continuing to hinder" his mission.
Turkey 'supporting terrorism'
As the diplomatic struggle to stop the Syrian crisis intensified, Damascus criticized Turkey's role in the uprising.
The Syrian foreign ministry said Ankara had played a "fundamental role" in supporting terrorism by opening its airport and border to al-Qaeda and other jihadists to carry out attacks inside Syria.
"The Turkish government has set up on its soil military offices where Israeli, American, Qatari and Saudi intelligence agencies direct the terrorists in their war on the Syrian people," the statement said.
Turkey has led the campaign against Syrian President Bashar al-Assad, hosting thousands of rebels fighters who have crossed the border to carry out attacks.
Inside Syria the fight for the second city of Aleppo continued on Thursday.
Rebels turned the gun of a captured tank against government forces on Thursday, shelling a military airbase used by war planes in the battle for Aleppo.
Assad's troops meanwhile bombarded the strategic Salaheddine district in Aleppo itself with tank and artillery fire supported by combat aircraft while rebels tried to consolidate their hold on areas they have seized.
In the capital Damascus, troops overran a suburb on Wednesday and killed at least 35 people, residents and activist organizations said.
The fighting for Syria's two biggest cities highlights the country's rapid slide into full-scale civil war 17 months on from the street protests that marked the start of the anti-Assad uprising.
The lightly armed insurgents are battling a well-equipped army that has overwhelming superiority on paper.
But the rebels have managed to capture some tanks and heavy weapons and their ranks are swelled by army defectors.
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