Continued fighting has raised questions about the viability of the USA strategy to end the war, which for the past year has focused on forcing the militants to the negotiating table, largely via more air strikes.
Taliban officials say they held "general discussions" with Washington's special envoy Zalmay Khalilzad, who is tasked with finding a negotiated end to Afghanistan's protracted and increasingly bloody war.
"Both sides spoke (about) an end to the occupation and a peaceful solution to the Afghan issue".
In the meantime, a source privy of the development has told The Wall Street Journal that Ambassador Khalilzad on Friday met with the Taliban political leaders in Doha, the capital of Qatar.
In July, Alice Wells, the US Principal Deputy Assistant Secretary for the Bureau of South and Central Asian Affairs met with Taliban officials to discuss ways to lay the groundwork for peace talks.
Both sides "agreed to continue such meetings".
Ghani thanked Khalilzad for his briefing and for his efforts and said that "we welcome any efforts to bring lasting peace in Afghanistan". The meeting was not confirmed on the record by either side.
Khalilzad's 11-day trip has included meetings with officials in Afghanistan, Pakistan, the United Arab Emirates (UAE), Qatar, and Saudi Arabia.
The meeting with Khalilzad and other USA officials took place on Friday and talks will continue, the Taliban said in a statement emailed to journalists. A Taliban official who was part of the four-person delegation to that meeting said it produced "very positive signals". Taliban officials want the withdrawal of American forces and a government in Kabul that reflects the group's view.
A US-led coalition forced the Taliban from power in 2001 in the aftermath of the September 11 attacks on Washington and NY.