Ghani said he had spoken late on Tuesday with US Secretary of State Mike Pompeo, who had stressed the importance of "ensuring the centrality of the Afghan government in the peace process".
Moscow had hosted a multilateral summit on Afghanistan last November, inviting officials from regional countries, besides a Taliban delegation.
Of note, for as much as he hypes up the US' possible loss of face following any prospective withdrawal from Afghanistan, Kakar doesn't talk about what a loss of face and money it would be for India if the Taliban seizes its many investments there in the aftermath. But in anticipation of such an order in the future, given the Trump administration's achievement of what it calls a "framework" for potential peace negotiations between the Taliban and the Afghan government, military planners have been considering plans for how a pullout might be conducted.
"The Americans told us (last month) that they would withdraw half of their troops from the beginning of February to the end of April", Taliban official Abdul Salam Hanafi was quoted as saying by RIA.
The Taliban, responding to Trump's speech, rejected any suggestion of a lingering USA focus on counter-terrorism after troops are drawn down, reiterating their long-held demand that all foreign troops get out.
Koofi urged the Taliban to "listen" to the Afghan people and to adapt to the current Afghan society.
Participants also agreed on the complete withdrawal of foreign troops from Afghanistan.
The Russian Embassy in Kabul said members of the Afghan diaspora in Russia organized the dialogue, though delegates privately acknowledged the host government played a major role in arranging it.
"The effort now is exactly with the goal to have an integrated peace settlement in which the withdrawal of troops will be one element", Atmar said when asked whether the meeting demanded complete withdrawal of forces by the USA and its allies.
The Taliban consider the Kabul administration a USA puppet but Ghani's allies in Washington insist Afghans should lead the peace process.
Stanikzai further demanded the removal of Western sanctions and travel bans on Taliban members, prisoner releases, an end to "propaganda" against the Taliban and clearance to open an official office.
Taliban official's comment came a day after President Trump told Americans his administration had accelerated talks for a political settlement in Afghanistan and would be able to reduce United States troops there as negotiations advance to end America's longest war.
The Taliban said the Moscow meeting - their most significant with Afghan politicians in recent memory - would discuss the withdrawal of foreign troops, peace terms and its vision for governance.
"When Mr. Karzai returns from Moscow we will see what he has brought. Let hundreds of such meetings be held, but these would only be paper (agreements) unless there is an agreement by the Afghan government; Afghanistan's national assembly and Afghanistan's legal institutions".
One outcome of reduced US leverage is a greater role for Moscow, which has worked behind the scenes to solidify influence in Afghanistan.
"They can go to school, they can go to universities, they can work", he added. The Taliban killed at least 47 security forces and policemen in separate attacks in the northern Kunduz, Baghlan and Samangan provinces over the past two days.