Mohsin Issa and Zuber Issa said the deal offered a “fantastic opportunity” to purchase a “ brand we have long admired”.
The group has also committed to keeping on Leon’s management team and staff.
Leon boss John Vincent, who co-founded the firm in 2004, said: “In some ways this is a sad day for me.”
“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,” he added.
Mr Vincent described the Issa brothers as “decent, hard-working business people” who would be “superb custodians” of the Leon brand.
He had previously said Leon was losing around £200,000 a week during lockdown.
“That probably means that versus what we would have been making, we’re losing probably an extra £800,000,” he said.
The deal includes 42 company-owned restaurants, as well as 29 franchise sites which are mainly found in airports and train stations across the UK and a handful of European countries such as the Netherlands and Spain.
Leon made sales of about £115m in 2019, running losses to fund its expansion within the UK and overseas before the pandemic stunted its growth plans.