The Taliban has in the past refused to negotiate directly with Kabul - a standing that does not appear to have changed - and was only prepared to talk with U.S. officials about the pull-out of foreign forces.
Afghan authorities have previously complained of being excluded from the discussions in Qatar, and warned that any deal between the U.S. and the Taliban would require Kabul's endorsement.
The militants have said they will only begin talks with the government once a firm date for the withdrawal of US troops has been agreed.
It is estimated about 15 million people - half the Afghan population - are living in areas either controlled by the Taliban or where the militants are openly present and regularly mount attacks.
US and Taliban officials have agreed to a framework for a peace deal in Afghanistan, with the USA touting "significant progress" at peace talks in Qatar over the past week.
The talks led by US Special Representative for Afghanistan Reconciliation Zalmay Khalilzad were aimed at finding a negotiated settlement to the 17-year conflict which has gripped Afghanistan following the fall of Taliban regime in 2001.
Khalilzad said in an interview with The New York Times that an agreement in principle was reached with the Taliban on the framework of a peace deal "which still has to be fleshed out" that will see the insurgents commit to guaranteeing that Afghan territory is not used as a "platform for worldwide terrorist groups or individuals".
Afghanistan said on Monday it has been reassured by Washington that progress in talks with the Taliban remains geared towards facilitating peace negotiations with the government in Kabul.
It also said that Ghani thanked Khalilzad and praised United States efforts to restart negotiations between the Afghan government and the Taliban.
"Either they join the great nation of Afghanistan with a united voice, or be the tool of foreign objectives", he told the militant group.
But Ghani called the U.S. -Taliban talks "part of our peace" and warned that a deal without Afghan involvement could lead to the kind of "disastrous" civil strife that followed the Soviet troop withdrawal from Afghanistan in 1989.
"We want peace, we want it fast but we want it with a plan", Ghani said in his address Monday.
The Taliban say they will only begin negotiations with the government once a firm date for troop withdrawal has been agreed.
"No Afghans want foreign forces in their country for the long term", Ghani said.
The Taliban have insisted on the withdrawal of foreign troops, and US President Donald Trump's clear eagerness to end America's longest war has weighed on the negotiations.
During the meeting with Ghani on Sunday evening, Khalilzad rejected media reports claiming he and the Taliban discussed the formation of an interim administration in Afghanistan, according to the statement.
"The Taliban have committed, to our satisfaction, to do what is necessary that would prevent Afghanistan from ever becoming a platform for worldwide terrorist groups or individuals", he told the New York Times.
In Monday's address to the nation, Ghani also stressed that U.S. and other foreign forces are in Afghanistan because they are needed here and that if there is to be any downsizing or pullout, the Kabul government will have to play a role in the talks. In a series of tweets on Saturday, Khalilzad announced he had briefed Afghan President Ashraf Ghani and that "nothing is agreed until everything is agreed".
Western diplomats describe Ghani as being in a tight spot, with no authority to decide on the troop departure and the final decision to declare the ceasefire resting with the Taliban. NATO's combat troops left Afghanistan at the end of 2014, but thousands remain in training, support and counter-terrorism roles.
Pelosi blocking Trump's speech
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