Sky News reports that Johnson has begun sounding out bankers about an auction of all or part of Bread Holdings.
The company includes Gail’s - which trades from around 50 sites - wholesale business Bread Factory and a number of subsidiaries.
A sale is expected to take place in 2020.
Speaking at a trade show earlier this month, Gail’s managing director Marta Pogroszewska said the chain was seeing “significant like for like growth” and looking to double in size to around 100 sites over the next three years.
Talk of a sale comes after Johnson’s reputation took a bruising with the collapse of the Patisserie Holdings, the café chain where he held an executive chairman role, earlier this year.
A major accounting fraud was discovered in the group's filings, which is currently under investigation by the Serious Fraud Office.
Johnson previously drew up plans to buy out Bread Holdings using equity from Patisserie Holdings in 2018, but the option is now off the table.
Gail’s was founded in Hampstead in 2005, and has been backed by Johnson since 2011.