In particular, Khalilzad told the New York Times that the two sides had reached an agreement in principle on a framework for a peace deal that would see foreign troops withdraw from Afghanistan over a period of 18 months.
The sides are working on a ceasefire that would lead to a sequenced agreement for dialogue, initially between United States and the Taliban and then between the Taliban and Afghan government, a source with knowledge of the talks told CNN last week.
The Taliban has repeatedly rejected the offer to talks with the Afghan government directly, saying no dialogue would be held with the Afghan government in the presence of foreign forces in the country.
Afghan President Ashraf Ghani assured his people that their rights would not be compromised in the name of peace with the Taliban.
"Our commitment is to provide peace and to prevent any possible disaster", Ghani said in an address to the nation.
According to the Associated Press, the president added, "There are values that are not disputable, such as national unity, national sovereignty and territorial integrity".
It said Khalilzad noted he discussed a ceasefire deal with the Taliban but there was no progress so far on the issue.
The US has about 14,000 troops in the country, most of which are present as part of a larger NATO-led mission to train, advise and assist Afghan forces.
US President Donald Trump´s clear eagerness to end America´s longest war has also weighed heavy on the discussions, and Ghani warned against rushing into a deal, citing violence in the aftermath of the Soviet withdrawal in 1989.
I missed it - Tiger happy to be back after shooting 70
Rose dedicated the victory to his caddie, Mark Fulcher, who had a heart procedure last week and was watching from home. On the par-3 16th, Rose holed a 30-foot birdie putt, right before Scott followed him in for birdie from 20 feet.
One of the key US demands is that the Taliban agree to keep Afghanistan from being used by militant organizations to launch attacks overseas, as Al-Qaeda did on 9/11. US officials accused the group's leaders of providing support to al-Qaida leader Osama bin Laden and his followers.
"We want peace quickly, we want it soon, but we want it with prudence", Ghani said.
The source added that the United States wouldn't announce any plans for a troop withdrawal without the Taliban entering into discussions with the Afghan government.
India is yet to respond to the latest dialogue that reportedly led to broad agreement on the principles of resolution of the conflict and withdrawal of the USA forces. No Afghan wants foreign troops to remain in their country indefinitely. "It is our policy to get to intra-Afghan talks", Khalilzad told reporters in the Afghan capital Monday.
Ghani said based on an global agreement, foreign troops will not be required in the future.
As well as the withdrawal of troops, the official said progress was made on addressing USA concerns that Afghanistan is not used as a base by al Qaeda or Islamic State for attacks against the United States and its allies.
The Taliban have in the past refused to negotiate directly with Kabul - a standing that does not appear to have changed.
On the issue of foreign troops in the country, he said no country wants this indefinitely but that now, Afghanistan needs these forces. Write to us in the Comments section, and visit our Facebook page.
Afghan security forces are already taking staggering losses, with 45,000 killed since late 2014, and morale is low.