The brothers’ petrol forecourt business EG Group has secured more than 70 restaurants across the UK and Europe.
In a statement issued on Sunday, EG Group said the deal – worth £10m according to the Financial Times – is part of “significant growth in its non-fuel and food service operations”.
The group also has a growing land portfolio, including the former Wellington bomber factory site at Blackpool airport which runs along Squires Gate Lane.
Leon chief executive John Vincent, who co-founded the chain in 2004, described a “sad day” but said that he has got to know the Issa brothers over the last few years.
He said: “They have been enthusiastic customers of Leon, going out of their way to eat here whenever they visit London.
“They are decent, hard-working business people who are committed to sustaining and further strengthening the values and culture that we have built at Leon, a business that has my dad’s name above the door.
“We have tried hard, done some good things, made a healthy amount of mistakes, and built a business that quite a few people are kind enough to say that they love.”
Mr Vincent founded the company with Henry Dimbleby and chef Allegra McEvedy with the brand focused on creating healthy fast food.
He said he is confident that under new ownership the brand will “flourish and have even greater appeal”, particularly outside London.
In a joint statement, the Issa brothers said: “Leon is a fantastic brand that we have long admired.
“As established entrepreneurs in the food service retail market ourselves, we have a huge admiration for the business that John and the Leon team have built over the years, and firmly believe that their culture and values closely align with our own.”
EG Group said it plans to open around 20 Leon sites a year from 2022.
The group already operates more than 700 food outlets in the UK and Ireland including branches for Starbucks, KFC, Burger King, Greggs and Subway.
It comes after the Issa brothers completed their £6.8bn acquisition of Asda in February, though the deal still needs to secure approval from UK competition regulators.