Lord Brodie is to chair the investigation, which has been set up by ministers to examine issues surrounding the Queen Elizabeth University Hospital in Glasgow and the delayed Royal Hospital for Children and Young People in Edinburgh.
He will meet Ms Freeman before the end of the year to discuss the timescale for the inquiry and the issues it will cover.
The under-pressure Health Secretary announced she had ordered a public inquiry into issues at the two hospitals in September.
This is in addition to an existing review into the design, build, commissioning and maintenance of the Queen Elizabeth University Hospital and Royal Hospital for Children, which is housed on the same site in Glasgow.
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Meanwhile, the opening of the Edinburgh children's hospital had to be delayed after Ms Freeman was alerted to safety issues there shortly before it was due to start receiving patients.
The Health Secretary said she had "announced this independent public inquiry following concerns, including from parents and families, over the quality of our NHS major infrastructure, its safety and compliance with standards and the impact that has on the delivery of healthcare to patients".
She described it as being "crucial work" and said she was "pleased that a person of Lord Brodie's stature and legal standing will lead this important inquiry".
Ms Freeman added: "The safety of patients and their families will always be my top priority - they must have the right support and information to give them confidence that they are receiving the best care possible from our NHS.
"This inquiry and its recommendations will help us learn lessons from recent issues so they are not repeated in the future.
"I have a statutory obligation to consult with the chair on the inquiry's terms of reference, and I also intend to share these with patients and families.
"I will provide a further update to Parliament early in the new year."