Shona Haslam, the Conservative leader of Scottish Borders Council, has said that she has already spoken to the British Government about the situation in Afghanistan following the country's takeover by the Taliban.
Cllr Haslam pointed to the fact that refugees from Syria had been rehomed in the district following the humanitarian crisis there.
A spokesperson for the authority said: "Scottish Borders Council has been engaging with the Scottish Government regarding our willingness to welcome refugees from Afghanistan. We are continuing to work with partners to ensure that we have everything in place to make this possible."
Dumfries and Galloway Council has not yet discussed the issue.
A Cumbria County Council spokesperson said:
They also pointed out that the county had become home 258 refugees since 2017 under the existing Government refugee resettlement schemes, which included four families from Afghanistan earlier this summer.
Under those schemes, the county has already pledged to resettle a further 137.