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.

Lt Col Gill Wilkinson was commanding officer of 154 (Scottish) Regiment Royal Logistic, based in Dunfermline.
She handed over to her husband, Lt Col Alan Wilkinson, last month.
The exchange makes the pair, who married in 1998, the highest ranking couple to follow each other in a command role.
Gill, 45, originally from Northern Ireland, was in the Army from the mid-90s but left when her two children were born.
She re-enlisted in the Army reserve in 2003 and took command of the regiment in 2014.
Husband Alan, 46, who was also in the Army before joining the Army Reserve, took on the role of main carer for their children when Gill became CO (commanding officer).
Gill told BBC Scotland: "It almost feels like I handed over the children to him two-and-a-half years before taking command and this feels very similar.
"You do feel a responsibility for a regiment. It is quite a parental role. Your soldiers are like your family.
"Normally when a commanding officer leaves command they don't look back at the regiment, they leave the next CO to carry on.
"Obviously I'm trying to do that but I can keep an eye out and ask how people are getting on and satisfy that parental bit."
Alan, originally from the Borders, met Gill at Dundee University before joining the Army.
He joined the reserves in 2009.
Alan said taking over from his wife was no different to any new job.
He said: "We have worked closely together throughout our career in the Army. I was in the fortunate position of being able to speak to Gill in the months before about it so it only took about four days to handover. It was fine."
Gill, who will now work with the Army Reserve part-time, said opportunities for women were "getting wider".
She said: "When I joined the Royal Logistic Corps it was because it offered, as far as I was concerned, the greatest opportunity to do the same job as men were doing.
"It is still majority male but the regiment I was commanding was about a quarter to a third female."
"I think ever since I joined the Army it has been increasing with more and more opportunities opening up.
"But we are really at the final bastion now where women are moving into combat roles."
Alan says his wife has offered him advice on his new role.
He says: "Some is good, some I listen to but I do make my own assessment."
A Scottish Borders quarry renowned for the quality of its stone could be reopened under plans to extract thousands of tonnes from the site.
[NEXT_CONCEPT]
A Scottish couple made an unusual bit of British military history when the wife handed over command of an Army unit to her husband.