Shrewsbury & Telford Hospital NHS Trust (SaTH), which manages both Princess Royal Hospital in Telford and Royal Shrewsbury Hospital, said it was looking into the incident, which took place earlier this month.
The trust has also confirmed that it has referred the case to the Information Commissioner.
The incident saw a total of 11 letters for patients sent to another patient awaiting an update on his own treatment.
All 11 letters were contained in one envelope, and included updates on different individual patients' medical treatment.
They appeared to be intended for one medical practice in Telford, but were instead sent to a man at a private address who was receiving information from the hospital about his own situation.
The letters were from the trauma and orthopaedic department of of PRH, and were all addressed to doctors at the Telford practice – detailing updates on conditions and treatment for individual patients who had been referred to the hospital.
Anna Milanec, director of governance and communications and senior information risk owner at SaTH, said the issue was being looked into.
"We can confirm that we have received the letters and we are carrying out an investigation," she said.
"We take our responsibility to maintain the confidentiality of patients, staff and others very seriously."
The letters were discovered by the partner of the man receiving a legitimate update on his treatment.
She did not want to be named but voiced her surprise at the discovery, and concern that people could be missing out on important details about their own health.
"We were really shocked," she said.
"It is quite alarming and obviously the people due to get these letters were not going to get them. It is just a bit of a shambles.
"I am concerned that the same thing could happen with a letter for my partner."