Morgan, who has more than 40 years' experience across the NHS, retired from full-time executive roles in the NHS in June 2024.
His most recent role was as group chief executive for Lincolnshire Community Health Services NHS Trust and United Lincolnshire Hospitals NHS Trust.
Morgan said: ‘I am passionate about the NHS, and I am eager for the two organisations, working together, to provide excellent integrated care to patients, both in our hospitals and in our communities. I believe this can be achieved by looking at how the trusts deliver modern responsive services that are safe, high quality and effective.'
Morgan will take up his new role on 1 October and his appointment will initially be for three years.
The trusts will remain separate statutory bodies.
Patricia Davies, chief executive of Shropshire Community Health NHS Trust, said: 'We have a long history of working collaboratively across the health and care system and I am delighted to be further supporting this approach with the appointment of Andrew as our joint chair-in-common.'
Jo Williams, chief executive of The Shrewsbury and Telford Hospital NHS Trust, added: ‘Andrew's wealth of NHS knowledge and leadership experience will be of huge benefit and will strengthen greater partnership working with our colleagues at ShropCom.'