US President Donald Trump last month declared talks with the insurgents "dead", citing a Taliban attack that killed an American soldier.
Negotiations had been in the final stages for a deal that would have seen the US pull troops from Afghanistan after 18 years in return for various Taliban guarantees.
But to the dismay of many Afghans and international observers, the deal included no immediate, comprehensive ceasefire, and rather would supposedly have paved the way for a reduction in violence and later talks between the Taliban and the Afghan government.
Roland Kobia, the EU special envoy for Afghanistan, said the talks' collapse provided a chance to push for a ceasefire which would, in turn, prove a large enough change in Afghanistan for Trump to consider resuming negotiations.
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"It's the right moment and the right opportunity to maybe go one step beyond a simple reduction in violence and explore ways in which a ceasefire... will take place," Kobia told Kabul journalists.
"The idea is really to see how we can move the ceasefire idea forward instead of leaving it for later... There is an opportunity here today."
In the meantime, violence in Afghanistan continues unabated. On Friday, at least 70 people were killed when a mosque in Nangarhar province was bombed.
When asked how the EU, which has only a limited footprint in Afghanistan, could leverage a truce, Kobia suggested that the Taliban might return to power in "one form or another" within months and so would entertain a truce to help normalise future relations with the European bloc.
"A ceasefire would be a token, a guarantee of goodwill and good preparation for the normalisation of these relationships," Kobia said.
The Taliban have steadfastly ruled out an immediate ceasefire, but last year downed weapons for a three-day truce.
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Afghanistan is currently in an uneasy waiting period following the first round of presidential elections on September 28.
Results were supposed to be released Saturday but have been indefinitely delayed due to "technical issues", the Independent Election Commission said.
Pierre Mayaudon, head of the EU delegation in Afghanistan, said a delay of a few days to finalise results was legitimate to ensure votes were fairly counted.
"But not many more days that again will go into weeks and will possibly raise the perception that something is happening," he told reporters.
US Defense Secretary Mark Esper meanwhile arrived in Kabul on Sunday on an unannounced visit to meet with US troops and Afghan President Ashraf Ghani.
"I'm looking... to get a really good feel for what's happening on the ground in Afghanistan, and to talk what the way ahead may look like as well," Esper told reporters accompanying him, according to a Pentagon transcript.
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"We think a political agreement is always the best way forward with regard to next steps in Afghanistan."
Currently, the United States has around 14,000 troops in Afghanistan.
Esper said that footprint could shrink to about 8,600 with no impact on counterterrorism operations against groups such as the Afghan affiliate of the militant Islamic State (IS) group.
Top US Democrat Nancy Pelosi, who is leading an impeachment probe into Trump, also made an unannounced visit to Afghanistan on Sunday as part of a bipartisan congressional delegation.
Pelosi, who has been bitterly critical of Trump's foreign policy, met with Ghani as well as US troops and diplomats.
Since Trump ended talks with the Taliban, Zalmay Khalilzad, the US special envoy leading the negotiations, has spoken informally with insurgents in Pakistan, raising the possibility that Washington is seeking to resume discussions.
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