The foes have spent much of the past week in a sixth round of talks in Doha aimed at ending America's longest war, but the Taliban say negotiations have become bogged down.
In a tweet, Taliban political spokesman Suhail Shaheen said the talks were pausing for the first day of the holy month of Ramazan but would be resumed on Tuesday.
'US-Taliban talks stumble over troop withdrawal'
Sultan Barakat, the director of the Center for Conflict and Humanitarian Studies in Doha, also said the talks would resume on Tuesday, and that they had made "good progress".
But Shaheen told AFP late Sunday negotiations were stumbling over the fundamental question of when foreign forces would depart Afghanistan.
Before the US agrees to any withdrawal, it is demanding the Taliban put in place security guarantees, a ceasefire and other commitments including an 'intra-Afghan' dialogue with the Kabul government and other Afghan representatives.
The insurgents insist they won't do any of these things until the US announces a withdrawal timeline.
At the end of a large peace summit in Kabul last week, President Ashraf Ghani offered the Taliban a ceasefire to begin on the first day of Ramazan, but the insurgents refused.
Suicide bomber leads Taliban attack on Afghan police HQ, 20 injured
Ghani on Monday reiterated his call for the Taliban to respect demands from last week's 'loya jirga' summit that saw thousands of tribal elders and Afghans meet in Kabul.
"Ramadan is a month of peace and reconciliation," Ghani said.
"I once again call on the Taliban to pay respect to this month and the demands of the people for peace and reconciliation reflected (in the Loya Jirga)."
On Sunday, at least 13 people were killed and dozens more wounded after a Taliban suicide bomber and several gunmen attacked a police headquarters in northern Afghanistan.
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