The sale of the supermarket chain to the Blackburn-based EG Group owners and TDR Capital is still awaiting official approval.
The Competitions and Markets Authority is looking into possible concerns over Asda petrol stations being included in the deal.
It is understood that the brothers have agreed to sell off 27 EG forecourts as a compromise.
Mohsin and Zuber Issa and TDR Capital said in a statement: “Over the course of the past 10 days, we have been working constructively with the CMA to offer remedies to address the CMA's competition concerns.
"Today, we are pleased to confirm that the CMA has indicated it has reasonable grounds to believe the proposed remedies are acceptable, enabling us to arrive at a conclusive outcome for the acquisition of Asda in Phase 1.
“As is usual in cases such as these, the CMA now has a period of 40 days to work through the detail of the proposed divestitures and therefore we are restricted in the level of information we are able to provide on specific sites.
"However, we have been comforted by the significant interest we have already received from potential buyers during this process, demonstrating the strong growth potential of our forecourts and the liquidity in the market.
"Over the coming months, we are confident that we will be able to agree a sale to suitable operators to take over all identified sites, and we will share more information in due course.”