Karzai had said on September 29 that he was ready to give militants a position in the government in exchange for peace.
Taliban spokesman Qari Yosuf Ahmadi responded to the president's overture with a reiteration that there can be no such talks until U.S. and NATO troops leave Afghanistan.
But Karzai spokesman Humayun Hamidzada said the spokesman's position is not shared by all Taliban militants.
Hamidzada told RFE/RL's Radio Free Afghanistan that a significant number of those fighters do not rule out laying down their weapons and being included in peace talks.
"The information we have received from tribal elders indicates that different groups operating inside Afghanistan under the Taliban name are discussing this issue seriously," Hamidzada said. "In this case, we don't expect huge developments in the very near future, but we hope that those who want peace and stability in Afghanistan will come step by step to join the ongoing peaceful process."
President Karzai has set at least two preconditions for peace talks. He has said he would negotiate only with Afghan Taliban -- not with foreign fighters -- and he has ruled out including militants with links to Al-Qaeda or other terrorist groups.
Karzai has also said he would personally go and talk to fugitive Taliban leader Mullah Mohammad Omar if he knew his whereabouts or "his phone number."
There was no response from Mullah Omar -- who is among the most-wanted men by U.S. authorities, with a multimillion-dollar bounty on his head.
There has been no official U.S. reaction to reports of Karzai's peace offer.
Tim Foxley is a guest researcher at the Stockholm International Peace Research Institute. In comments to RFE/RL, Foxley speculates that the president's willingness to engage extremist leaders like Mullah Omar will not be welcomed by his Western supporters -- particularly the United States.
"These are not people that, certainly, the American government would have any interest in talking to," Foxley says. "So I think it would risk making a split between Karzai and his Western allies."
Some of Karzai's other international supporters appear to have accepted the Taliban presence as a harsh reality, and indicated their willingness to back Karzai's diplomacy with them.
British Foreign Minister Des Browne has suggested that the Taliban will have to be involved in the country's peace process. Browne said the Taliban "are not going away more than I suspect Hamas are going away from Palestine."
Hamidzada said that during Karzai's trip to the UN General Assembly in New York last week, UN Secretary-General Ban Ki-moon and representatives of many countries sought a comprehensive strategy -- involving both military and diplomatic components -- in dealing with the Taliban.
Foxley says that -- in theory -- talking to moderate Taliban and separating them from hard-core fundamentalists and Al-Qaeda supporters would weaken the insurgency.
But he acknowledges that it is not easy to identify "moderate Taliban" and find their partner for discussion.
Even Karzai has suggested that his government has had trouble finding a proper channel of communication with the Taliban.
"We are ready to negotiate to bring peace [to] this country," Karzai said. "Continuation of the war, explosions, and suicide attacks should be stopped in any way possible. There were some contacts with [Taliban] in the past. But there is no specific, clear-cut line of communication -- I mean, there is no official place for communication with the Taliban. I wish there were such a place."
So as Karzai sends out trial balloons for peace talks, the question remains as to how authorities will verify the authenticity -- and firmness -- of the responses.