In a statement on its website the restaurant team said they were selling up to focus on the farming and cider growing side of the business.
They hope the sale will take place this summer, with the site trading as normal until then.
Birch's owners Sam Leach and Beccy Massey started out running pop-up supperclubs around the city before launching a permanent restaurant in 2014.
As well as winning favour with local residents, Birch received a glowing review from Marina O’Loughlin in The Guardian, who was left scratching her head ‘really hard for something to whinge about’.
It was also recommended in the 2018 Michelin Guide Great Britain and Ireland, and was ranked the 37th best restaurant in the UK at Restaurant magazine's 2016 National Restaurant Awards.
A statement on the Birch website said: “Over the past year we have suddenly and unexpectedly been faced with some fantastic opportunities which we felt we could not turn down: to rent an ancient orchard in Chew Magna, to rent a barn in Failand, and to take on some more land for vegetable growing in Whitchurch.
“We always had long term plans to move into farming, and the time now seems right…sometimes you have got to grab the fish by the tail, and we have just grabbed it.”
However, the Birch team said the business was still trading well.
“This isn’t the bubble bursting on the Bristol Restaurant scene, the Bristol restaurant scene has never been in such health,” they added.
“We are leaving a profitable business and putting it up for sale to another operator, most likely an owner operator, who wants to buy a fully functional, well equipped and well maintained restaurant.”
Bristol currently has 37 restaurants recommended in the Michelin Guide, four with one Michelin-star and two with Bib Gourmands.