Dozens of unarmed members of the outlawed organisation entered Kabul on Saturday where they exchanged hugs with troops and made selfies. He also said that a cease-fire could be accompanied with visits to their prisoners and treatment for their fighters at hospitals in Afghanistan.
This is the first time that the Taliban has ever extended an offer of this kind to the Afghan government.
The attack came as President Ashraf Ghani announced an extension of the government's week-long ceasefire with the militant group after both sides agreed to halt hostilities for Eid.
Afghan government has said it will continue operations against other insurgent groups, including Daesh, and only Taliban are part of the ceasefire.
"If their leaders come, sit and talk just like their soldiers we will have peace tomorrow", Said Hasibullah posted on Facebook under a photo purportedly showing a Taliban fighter and Afghan soldier having a cup of tea together.
There were also rare scenes of impromptu diplomacy as the interior minister, Wais Barmak, met Taliban members on the outskirts of the capital and district governors broke bread with their Taliban "shadow" counterparts over lunch.
The Taliban had "exploited" the opportunity to show their popularity among ordinary Afghans, a Western diplomat in Kabul told AFP.
Men and women gathered around the soldiers and Taliban fighters and urged them to keep their weapons holstered before they hugged each other. Some of our members went there to celebrate Eid.
A suicide bomb attack at a cease-fire gathering in eastern Afghanistan Saturday killed a number of people, majority Taliban fighters, according to reports.
"(That's) no bad thing if they are able to see the benefits of talking not fighting", he said.
Barnett R Rubin, an Afghanistan at the Center on worldwide cCooperation, said the ceasefire will make it easier for high-level talks to occur as Taliban fighters had strictly obeyed the command of their leaders.
US President Donald Trump unveiled a new strategy for Afghanistan past year, vowing to deploy more troops, on top of the 11,000 already in the country, to train and advise Afghan security forces.
Since the beginning of the year, Taliban and Islamic State militants have staged 13 attacks in Kabul city, killing more than 300 people and wounding more than 500 others.
"We want all the parties to extend the ceasefire", Obaidullah, a resident of Nangarhar said.
They said they would continue attacking US-led North Atlantic Treaty Organisation troops.
A video showed a huge crowd of people screaming and whistling as they welcomed the Taliban.
"I am here to offer greetings to our brothers in the police and army", Taliban commander Baba said.
There was no immediate claim of responsibility for the attack, but the Taliban said it was not responsible.
But as Mr Ghani was making his address, a suicide vehicle bomber blew himself up in a crowd of Taliban, security forces and civilians celebrating the ceasefire.