Members of Scottish Borders Council are due to debate the issue tomorrow, with plans already under way to house the textile artwork in a new building to be constructed next to the Tweedbank station.
The tapestry, the brainchild of Alexander McCall Smith, depicts 420 million years of Scottish history and was hand-stitched by 1,000 volunteers over two years. It is currently on tour around the country.
Councillor David Parker, leader of Scottish Borders Council, said: "The Great Tapestry of Scotland is a very special and unique work of art.
"Locating it in Tweedbank in a purpose-built facility will truly showcase this magnificent tapestry. Its location at the Tweedbank station will afford many visitors a fantastic opportunity to take the train to the tapestry and also to view other Borders attractions such as Abbotsford, Bowhill and the Borders abbeys.
"We are at an early stage of the project's development but I can see no reason why we can't deliver something very special for the Scottish Borders."
McCall Smith, co-chairman of the Tapestry Trust, added: "The tapestry has already brought immense pleasure to the many thousands of people who have seen it; now it faces a bright future in a lovely setting."