The two companies agreed an all-share merger last month to create a group worth about £1.9b with a fleet of around 40,000 vehicles and a workforce of 70,000 people.
The deal is effectively a takeover by National Express and will see its shareholders own around 75% of the combined group, with around 25% held by Stagecoach shareholders.
The merger, which will be voted on by shareholders, values Perth-based Stagecoach at around £437million.
Stagecoach, which was founded by brother-and-sister Ann Gloag and Brian Souter, a former SNP donor, said the CMA's decision to step in will delay the planned sale of parts of its inter-city coach businesses to ComfortDelGro Corporation Limited.
It added that the CMA wants to "maintain the businesses in their current shape" as it prepares to launch an initial investigation.
But it said the merger partners continue to believe the coach sell-off will be a "comprehensive solution to any competition concerns that might arise from their overlapping coach operations".
It said the firms "will engage with the CMA to allow the Stagecoach coach disposal to complete as soon as possible".
Stagecoach revealed deals to offload the marketing, retail and customer service operations of Megabus UK and the South West Falcon coach service last month in a bid to appease expected competition concerns, as well as its 35% stake in the Scottish Citylink Coaches joint venture.
Stagecoach said the CMA's order would not "materially affect the day-to-day operations of either National Express or Stagecoach, and the parties will continue to work with the CMA in relation to its review of the combination".
It went on: "At this stage, the boards of National Express and Stagecoach continue to expect the combination to complete around the end of 2022."
The merger comes as both firms have been hit hard by the pandemic, with passenger numbers slumping due to lockdowns, remote working and a switch away from public transport.
The two companies outlined plans to save at least £45m in annual costs following the merger. This is set to see around 50 roles cut across their head offices, IT and corporate departments, as well as some overlapping senior management positions.
The merger follows a previous attempt at a deal in 2009, when National Express rejected a £1.7b approach by Stagecoach.
National Express has bus and coach networks across the UK and Spain, while it also runs school bus services in America and a rail franchise in Germany.
Stagecoach is UK-focused and is Britain's biggest bus and coach operator.