The landmark meeting will come in the coming weeks, before the establishment of a Taliban office in Qatar, according to Western and Afghan officials.
The Taliban have refused previously to recognise the government of President Hamid Karzai.
Senior officials in Kabul say the Taliban have agreed to the meeting.
The militant group, contacted by the BBC, refused to comment on the move.
The Taliban have so far insisted they would only talk to the US and other allies of the Kabul government.
A senior Afghan government official told the BBC: “Even if the Taliban office is established in Qatar, we will obviously pursue other efforts in the region, including Saudi Arabia and Turkey.”
He continued: “Saudi Arabia has played an important role in the past. We value that and look forward to continued support and contact with Saudi Arabia in continuing the peace process.”
President Karzai was angered by US and Qatari efforts to kick-start the peace process without consulting his government fully.