Under the terms of the possible all-share tie-up, National Express would own around 75% of the combined group and Stagecoach shareholders around 25%.
It comes after both firms have been hit hard by the pandemic, with passenger numbers slumping during the crisis.
The groups have outlined plans to slash costs as part of the potential merger, with National Express saying it expects to find annual savings of at least £35 million, with around 25% by the end of the first year.
If the talks lead to a deal, the combined group would see Stagecoach chairman Ray O’Toole become chairman of the board.
National Express boss Ignacio Garat would be chief executive of the enlarged group.Internet Explorer Channel Network