The Great Tapestry of Scotland is a scheme to stitch "the entire story of Scotland" through a 143-metre long tapestry.
The Scottish Government has pledged to commit up to £2.5 million towards the projects delivery while the council will spend up to £3.5 million.
Councillors are now to consider a private report next week on two potential locations for the projects visitor centre in Tweedbank and Galashiels.
Members will hear both the previously identified Tweedbank site, opposite Tweedbank Railway Station, and a town centre site in Galashiels are viable options for the siting of the visitor centre.
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Council Leader David Parker said: "We are proposing that we proceed to look in further detail at the Galashiels option as this may attract further additional funding that the Tweedbank project could not access and it would also act as a significant town centre regeneration project.
"Siting the visitor centre in Galashiels town centre would have the potential to unlock substantial benefits for the town, including transforming it into a true visitor destination, which could encourage further positive developments in the town over a number of years.
"The report before Members demonstrates that both sites are viable and that the Tweedbank site has a very strong business case.
"Councillors are aware there is very strong support for the Galashiels project from Energise Gala, town centre traders and a whole variety of other business and community interests and I am very confident that there will be a happy ending for the Tapestry project in the Borders."
If a site is approved, a further report setting out more project details, including potential phasing and funding sources, will be brought before the local authority in November.
(LM/CD)
