According to Zabihullah Mujahid said the talks, which had been taking place in Doha, finished after midnight.
An agreement would allow President Donald Trump to achieve his aim of ending Washington's longest-running war, launched after the September 11, 2001 attacks on the US..
President Ashraf Ghani spoke during the Muslim holiday of Eid-al-Adha and as USA and Taliban negotiators continue their work in the Gulf nation of Qatar, where the insurgents have a political office.
"Our future can not be decided outside, he said, adding: "The fate of Afghanistan will be decided here in this homeland".
Afghanistan's president on Sunday rejected foreign interference as the United States and the Taliban appear to be closing in on a peace deal without the Afghan government at the table.
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FILE - Special Representative for Afghanistan Reconciliation Zalmay Khalilzad at the U.S. Institute of Peace, in Washington, Feb. 8, 2019.
"We want peace and we want participation in the peace process".
"Bloodshed should stop, and peace should be established in Afghanistan", said Khan Mohammad, a Wardak resident.
The Taliban do not recognise the Kabul government and have refused to talk to it.
The Taliban stage near-daily attacks across Afghanistan, mainly targeting Afghan forces and government officials but also killing many civilians.
Eid al-Adha 2019
In pic: A Kenyan child attends prayers to mark the Muslim holiday of Eid al-Adha at an open field in Nairobi, Kenya. Muslims around the world celebrate "Eid-al-adha" with a feast and by reconnecting with their friends and families.
The Taliban control about half of Afghanistan and are now at their strongest since the US-led invasion toppled their five-year-old government in 2001.
Washington has said it wants a peace deal finalized by September 1, and some US officials have hinted at the possibility that the Afghan presidential election set for September 28 could be canceled in the event of a peace settlement and the formation of an interim government that the Taliban would join.
However, both Daesh's affiliate and al Qaeda remain active in the country.
Addressing the nation after performing Eid al-Adha prayers at the Presidential Palace, the Ghani stated that "Afghans want to embrace lasting and national peace" that could ensure the honour and dignity of the people, reports Xinhua news agency.
"Leaders on all sides of the war in Afghanistan must take this to heart as we strive for peace", he said. More than 2,400 US service members have died in Afghanistan since then.
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The US and North Atlantic Treaty Organisation formally concluded their combat mission in Afghanistan in 2014.