Some Taliban militants are reportedly disillusioned with fighting
Plans to engage the Taliban in talks are gathering pace, with Afghanistan inviting members to a peace council and reports of a secret UN meeting.
At a key Afghanistan summit in London, President Hamid Karzai vowed to reach out to "disenchanted brothers".
A UN official told news agencies that UN special envoy Kai Eide had met Taliban members in Dubai on 8 January after they had asked for talks.
Mr Eide denied meeting on that date but refused to comment on other dates.
The one-day conference in London saw world leaders pledge $140m (£87m) to win over low-level Taliban fighters.
Delegates also said that Afghan forces could take control of security in some provinces by the end of 2010 and that the process could be complete in five years.
'Talks about talks'
At the summit, Mr Karzai again urged the Taliban to renounce violence.
By Lyse Doucet, BBC News
Whether or not the report of UN talks is true, it's clear that anyone who mediates in this process would want to keep it secret. Taliban leaders who reach out face great danger. They would need to know they would be protected. The reports speak of talks about talks, underlining how this process is still at a very early, very delicate stage.
A number of military and diplomatic sources in Afghanistan say they are hearing reports that some senior Taliban are tired of fighting, and would be interested in a political solution.
But in a situation where intelligence has often proved to be dangerously faulty, it's very hard to get the measure of what the Taliban is thinking and whether indeed this movement is divided and capable of being split from al-Qaeda and more hardline elements as Hamid Karzai and his allies want to believe.
"We must reach out to all of our countrymen, especially our disenchanted brothers, who are not part of al-Qaeda, or other terrorist networks, who accept the Afghan constitution," he told delegates.
He invited moderate Taliban to a peace summit.
The UN official, who would not be named, said that Mr Eide had met members of Taliban's Quetta Shura - senior commanders known to be based in the southern Pakistani city.
"They requested a meeting to talk about talks. They want protection, to be able to come out in public," the official was quoted as saying by Reuters.
Speaking to the BBC, Mr Eide, the UN Special Representative, categorically denied any talks took place on 8 January, saying he was only in Dubai for several hours en route to Kabul.
As for other dates, Mr Eide said he never commented on such reports, whether true or false.
Taliban spokesmen have repeatedly said they have no interest in talking to Mr Karzai's government.
However, Reuters quoted one Taliban spokesman, Qari Mohammad Yousuf, as saying: "I cannot say a word regarding these peace talks. The Taliban leadership will soon decide whether to take part in these peace talks."
The BBC's Lyse Doucet says that, if true, the report of secret talks would be a significant development.
The final communique from the summit in London said it welcomed Afghanistan's goal of taking charge of the "majority of operations in the insecure areas of Afghanistan within three years and taking responsibility for physical security within five years".
COMMUNIQUE'S KEY POINTS
Handover security duties in Afghan provinces starting in late 2010 or early 2011
Funds to reintegrate Taliban who cut ties with al-Qaeda
Hold a 2010 summit in Kabul to develop concrete plans for the Afghan government programme
Backs start of discussions on a new Afghan-led IMF programme
Increase share of aid delivered through the Afghan government to 50% in two years
Increase Afghan military strength to 171,600 and police numbers to 134,000 by October 2011
UK Foreign Secretary David Miliband told the BBC that 2010 would be a "decisive year" for Afghanistan.
"The insurgency needs to be reversed, the promises of the Afghan government need to be turned into reality... and the commitment of the international community to the military effort, to the civilian effort and to supporting a political settlement, needs to be put in place," he said.
He added: "The lynchpin is the Afghan people, because they are the greatest resource against the insurgency."
The summit said the Afghan government had acknowledged it had to tackle corruption.
US Secretary of State Hillary Clinton said reforms planned by President Karzai, such as tackling corruption and effectively managing aid, were important and the US would be watching them carefully.
The summit said the Afghan government had made progress on economic development, and it hoped it would continue to boost agriculture, human resources and infrastructure.
The BBC's diplomatic correspondent Jonathan Marcus said the theme of the conference was unity and coherence, but also an acknowledgement that there was no military solution to Afghanistan's problems.