Tilbury Douglas is understood to be on board to deliver both projects together valued at nearly £60m.
Bulldozing Arrowe Park Hospital is expected to take about five weeks, allowing Tilbury Douglas to start main construction in September and the new facility is scheduled to open in March 2024.
The job starts the first phase of the Urgent and Emergency Care Upgrade Programme at Arrowe Park.
At the same time, Stockport NHS Foundation Trust has also revealed it has gained planning for a £30m campus at Stepping Hill Hospital to significantly improve emergency and urgent care services.
Tilbury Douglas is expected to start work on the extension at the Hazel Grove site, which will increase the size of the current clinical area by more than 300 sq m, later this year this year.
The project involves a 12,000 sq ft extension to the A&E unit will result in a new children’s emergency department and consultation areas, as well as a larger ambulance and patient reception.