The tie-up, which would value Stagecoach at about 445 million pounds ($609 million), comes as government support to help the companies get through the pandemic is due to expire.
National Express has bus and coach operations in Spain and Britain, runs school buses in the United States, and has a German rail contract.
Stagecoach, solely focused on Britain, is the country's biggest bus and coach operator. The company was founded by Brian Souter and his sister Ann Gloag in Perth in 1980, starting out with just two buses bought with their father's redundancy money.
They still own stakes in the company - Souter has about 14.55% and Gloag around 10.47%, based on Refinitiv data.
Under the terms of the potential takeover, Stagecoach shareholders would receive 0.36 new National Express shares for each Stagecoach share, giving them a 25% stake in the merged group.
That represents an 18% premium on the closing price of Stagecoach's shares on Monday.
Stagecoach's stock jumped 20% to 81 pence in early trading on Tuesday. National Express rose 7% to 239 pence per share, giving it a market value of 1.48 billion pounds.
During the pandemic, both Stagecoach and National Express received government support to keep services running when passenger numbers were down, but that funding will end in the coming months.
Transport usage is starting to recover as workers return to offices, but is still not near its pre-pandemic levels.
The companies said the tie-up would make pre-tax cost savings of 35 million pounds per year through efficiencies such as National Express utilising Stagecoach's depots for its coach operations.
Stagecoach runs the megabus service, which like National Express, operates coach services between British cities.
Citi analysts said the deal would be positive for the transport sector. Rival UK operators FirstGroup and Go-Ahead also rose around 3% on news of the potential deal.
"We have argued for consolidation in the UK Public Transport given the structural challenges facing the industry, i.e. the pandemic impact on passenger volume and capex headwinds related to the adoption of environmentally-friendly vehicles," Citi said in a note.
The companies said that discussions and due diligence remained ongoing and there could be no certainty of a formal offer.
Should a deal go ahead, Stagecoach's chairman Ray O'Toole, a former chief operating officer at National Express, would become chair of the combined group, while National Express chief executive Ignacio Garat would continue as CEO of the merged entity. ($1 = 0.7310 pounds)
(Reporting by Sarah Young; Editing by Kate Holton, Michael Holden and Jane Merriman)
By Sarah Young