Announcing the closure on Instagram, Kenny said he hoped the restaurant would live on and be able to find a new home, but added that he and team may also look at pursuing new projects in the future.
Located within the Artist Residence hotel on Regency Square, The Set was opened by Kenny in 2015.
In 2017 the chef also took over responsibility for the Artist Residence's breakfast offering, and then in 2018 was put in charge of the hotel's entire F&B operation.
Kenny has also confirmed that he would be leaving his role overseeing the F&B at Artist Residence when The Set closes on 31 March.
He said: "After almost six years residing and working alongside Artist Residence this chapter is coming to an end.
"We will be leaving our residence at the hotel on 31 March 2021. We want to close the chapter by putting out the best food we have ever cooked, so please do book in to come and say goodbye with a final meal with us.
"Big thanks to everyone who has been a part of the restaurant over the years. Staff, suppliers, friends, family, collaborators and the HMRC phone line.
"There are too many people to mention but you know who you are. Again a huge thanks; your help, support and compassion has fuelled the restaurant and made it something we are proud of.
"It’s been a true pleasure cooking for our customers and an honour to welcome you to the pass. To those who have become regulars and friends to us, we hope to share a drink with you one way or another soon.
"We hope The Set will live on. The new year may bring us a new home, or we may look at new projects such as pop-ups, exclusive private dining and some other stuff (once we think of it). We shall keep you posted."
Earlier this year Kenny closed his neighbourhood-style cocktail bar The Fix, which operated alongside The Set at Artist Residence.
He has also recently launched Five Star Kebabs in partnership with fellow The Set chef Luka Dmitryi, which initially operated on collection-only basis from the window of a domestic kitchen.