The burger chain is looking to agree a company voluntary arrangement (CVA) to help put them on firmer financial ground by securing hefty discounts on rental costs.
The chain has branch in Leeds city centre as well as in Harrogate and
A CVA allows a business to close loss-making stores and was recently used by embattled toy retailer Toys R Us.
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Accountancy giant KPMG, which is handling the process, said Byron's directors were using the move in order to secure the company's future amid "gathering economic headwinds".
However, it would need to win the backing of landlords and creditors before pushing forward with the process.
The announcement comes as the group confirmed that Three Hills Capital Partners would strike a deal to become the biggest shareholder in the group as part of a sale process linked to the CVA.
Will Wright, restructuring partner at KPMG, said: "Over the last 10 years, Byron has grown to become a stand-out name within the UK's casual dining sector.
"However, in recent times, certain parts of its portfolio have not met expectations, and with gathering economic headwinds starting to impact the sector more profoundly, the directors embarked upon a strategic review of the business as a means of safeguarding its long-term future."
KPMG added that no restaurants will close on day one, and employees, suppliers and business rates will continue to be paid on time and in full.
Mr Wright added: "As part of this strategic review, the directors have been successful in negotiating a financial restructuring with the company's lenders and shareholders, which will enable new investment to come into the business.
"Completion of this financial restructuring is conditional on the approval of today's CVA proposal, which is designed to tackle the cost of the company's leasehold obligations across its UK restaurant portfolio.
"As with similar CVAs, this arrangement seeks to strike a balance which provides a fair compromise to landlords, while allowing the viable part of the business to move forward across a smaller, more profitable core estate."
Byron, which was founded by Tom Byng in 2007, employs around 1,800 people and has 67 leasehold restaurants and nine non-operational leasehold sites across the UK, including its head office in London.
Simon Cope, Byron chief executive, said: "Byron's core restaurant business and brand remain strong but the market that we operate in has changed profoundly.
"In order to continue serving our loyal customer base, we need to make some critical and difficult changes to the size and shape of our estate.
"With the support of our new owners, our creditors, landlords and other business partners, I'm confident Byron will able to continue providing our consumers with the best burger experience.
"The teams in our restaurants are always such an inspiration and we will work hard to support them throughout this difficult process."
Here is a list of the 20 restaurants at risk of closure from Byron's restructuring proposal.
:: Gateshead Metro Centre
:: Hoxton Square
:: Manchester Corn Exchange
:: Manchester Deansgate
:: Store Street, London
:: Stratford upon Avon
:: Wandsworth, London
:: Westbourne Grove, London