The restaurant, which has been run by head chef Billy Boyter since 2013, announced the news late last month on social media.
Posting on X, Boyter wrote: "After a lot of thought and consideration, my family and I have decided it's the right time for us to sell The Cellar. It's been an amazing 10 years, and I'm so proud of what we've created in our time here.
"I'd like to thank all of our amazing suppliers. We could have never achieved what we did without you. It's been an absolute pleasure working with you all. And, of course, thank you to all of our amazing guests, some of whom we now regard more as friends. We'll miss all of you. Thank you so much for being part of the journey."
Boyter, who was previously head chef at Michelin-starred Number One in Edinburgh, took over the restaurant in 2013 following the death of Scottish chef Peter Jukes.
He re-opened it in 2014 with the restaurant being awarded a Michelin star the following year, which it has retained ever since.
The restaurant is being sold by Shepherd Chartered Surveyors for offers upwards of £500,000.