There are former presidential candidates, Abdullah Abdullah and Ali Ahmed Jalali — both of whom were said to be Washington's preferred candidates in the last presidential election in 2009. There's also Rashid Dostum, who leads the minority Uzbek ethnic group and Mohammed Mohaqiq, the leader of another minority ethnic group called the Hazaras.
Also in the group is Ahmed Zia Massoud, a former Afghan vice president and the brother of anti-Taliban fighter Ahmed Shah Massoud, the charismatic leader of the ethnic minority Tajiks who died in an al-Qaida suicide attack two days before the Sept. 11 attacks that provoked the U.S. invasion of Afghanistan in late 2001.
A senior official with Hekmatyar, who is familiar with the many negotiating channels of his organization, confirmed that representatives have met with the opposition. He said the talks were nascent, but refused to give additional details.
Taliban spokesman Zabiullah Mujahid denied that the Taliban were talking with the opposition group. But a second Taliban official confirmed that the Taliban has been in contact with opposition members in Kabul.
All the officials spoke on condition of anonymity because of the sensitivity of the negotiations.
Gailani, the founder of the united opposition group, said the group was in discussions with Taliban interlocutors close to Omar.
The opposition expects to field a consensus candidate in next year's presidential election. Reaching an understanding with both the Taliban and Hekmatyar's group would give them a greater chance to win in the 2014 polls and cobble together a multi-party government that could bring stability to a post-Karzai government.
"It's no secret that almost nobody wants any version of the Karzai clan to remain in power," said Anatol Lieven, a professor in the War Studies Department at King's College in Britain who has met with current and former members of the Taliban.
But he said that he did not think that the opposition's talks with the militant groups would yield anything soon.
"They are part of a gradual process of building up links and exchanging views, which could lay the foundation for a settlement later, though only after Karzai has departed the scene," Lieven said.
Hekmatyar has laid out a 15-point plan for Afghanistan's future that calls for a broad-based government, nationwide elections, an interim administration and a series of election reforms.
The Taliban have been less clear about their vision of a future Afghanistan. However, late last year Omar, the one-eyed, reclusive leader, issued a statement in English that seemed unusually conciliatory.
"As to the future political destiny of the country, I would like to repeat that we are neither thinking of monopolizing power nor intend to spark domestic war," Omar said.
"The future political fate of the country must be determined by the Afghans themselves without any interference from big countries and neighbors, and it must be Islamic and Afghan in form," he added.
Talks with the U.S. were temporarily scuttled in early 2011 by Afghan officials who were worried that the secret, independent talks would undercut Karzai. They quietly resumed with each side seeking small signs of cooperation, but the Taliban shut down all talks with the United States after it refused to release their colleagues from Guantanamo Bay.
A senior U.S. official said there have been "no, no, no direct contacts of the U.S. with Taliban since January 2012."
Apparently frustrated by the lack of any progress in talks with the U.S., two Taliban officials told the AP that the movement's governing council was contemplating removing Tayyab Aga — special assistant to Omar during the Taliban's rule — as their lead negotiator because he "could not achieve the expected results."
Mullah Abbas Akhund, the Taliban's health minister, is being tapped as Aga's replacement, according to the two Taliban officials.
Kathy Gannon is AP Special Regional Correspondent for Afghanistan and Pakistan and can be followed on www.twitter.com/kathygannon