The local authority had previously offered to rehome four Afghan families before the recent events where the Taliban took over the country.
Up to 20,000 Afghan refugees could now seek asylum in the UK and numerous councils in Scotland and south of the border have already stepped forward to offer shelter.
Last week, the leader of Scottish Borders Council said she was already in contact with the British Government about bringing refugees to the area. Cumbria County Council has also said it is "ready to help".
Elaine Murray, the leader of Dumfries and Galloway Council, said:
Dr Murray added that the district had become home for 20 families from Syria since 2016.
Scottish Secretary and Dumfries and Galloway MP Alister Jack hoped the local authority would play a part in accommodating those fleeing Afghanistan.
He said: "The current, and previous councils, in the region have a good record in providing homes for a number of refugee families and individuals during past international emergencies.
The House of Commons was packed on Wednesday as Parliament was recalled to discuss the crisis in Afghanistan.
The seven-and-a-half-hour emergency debate heard how hundreds of thousands of innocent people had been driven out of their homes on fear of being killed by the Taliban.
Among them are Afghans who worked with the British Army as translators and in other support position for the allied forces, and these people and their families are amongst those most at risk.