Berwickshire-based Hutton Stone is behind the bid to bring the Swinton Quarter Quarry back into use.
It says the rock there has "very good qualities" and some of it could be used on Historic Scotland properties.
The firm has made scoping request to Scottish Borders Council which could lead to a full planning application.
The proposals would see an average of 1,000 tonnes of stone extracted over a period of about 20 years.
Scottish Borders Council's planning department is currently considering the scoping request.
A Scottish Borders quarry renowned for the quality of its stone could be reopened under plans to extract thousands of tonnes from the site.