Both the US and Germany have been pushing for such a representation in an effort to kick-start negotiations.
The office is seen by some as a key step towards ending the 10-year-long conflict in Afghanistan.
But it still remains unclear if the insurgents, who claim to be winning the war, are prepared to engage in truly meaningful peace talks - and whether they could take place while international forces continue to kill Taliban fighters and commanders, says the BBC's Quentin Sommerville in Kabul.
Importance of address In their statement, the insurgents said Afghanistan's "current problem" began with the US-led invasion of 2001 and "the two main sides which were involved in this are the Islamic Emirate of Afghanistan [Taliban] and on the other side is the United States and their foreign allies".
AnalysisThe establishment of an office is thought by some to be a critical step in reaching a political settlement to the 10-year long conflict. It would give the group an address where negotiators could meet. Establishing the authenticity of would-be negotiators from the Taliban has been a problem in the past.
It is the US and Germany that have been pushing for this. Earlier preconditions that the insurgents would have to lay down arms before any such representation appear to have been dropped. The push for a peace process, with a reluctant President Karzai falling in line, appears to be under way.
Some senior military commanders here say that the Bonn conference, where the international community gave a long-term commitment to Afghanistan, was a wake-up call for the insurgents. They face the prospect of growing old, as exiles, in the Pakistan city of Quetta, commented one senior Isaf commander.
But it is far from certain that the Taliban truly want to negotiate. They know foreign troops are leaving in 2014. And there will be reluctance from some within the group's leadership to sit down and talk with representatives from countries who are killing Taliban soldiers and commanders.
The statement added that they wanted prisoners released from the US-run detention centre in Guantanamo Bay, as part of a deal.
And it also rejected as false reports by "Western press and officials" about negotiations.
The agreement to set up the liaison office would give the group an address where negotiators could meet, says our Kabul correspondent, and some see this agreement as a critical step.
Establishing the authenticity of would-be negotiators from the Taliban has been a problem in the past.
Last month, President Karzai gave his first public support to the plan - having previously rejected the idea, angry that the US and Germany had discussed potential locations without him.
Kabul has repeatedly stressed that it will not accept any foreign intervention in negotiations with the Taliban.
Efforts to hold talks have been hit by a string of setbacks, including the assassination in September of Burhannudin Rabbani, the head of the Afghan High Peace Council which had been liaising with the militant group.
The Taliban denied being responsible, but the attack added to the sense of mistrust.
US and Afghan officials have also stressed that Pakistan - where the Taliban's leadership are believed to be based - must be involved in the process.