The interview, when it was broadcast on Sunday, prompted a backlash from relatives who said they had heard no such apology at the meeting.
In a statement they said: “Despite the gravity of the discussions, at no point during the session was a formal apology offered by either the first minister or the cabinet secretary.
“The meeting was intended to provide a platform for those affected to seek transparency and acknowledgment of the hardships endured.”
While welcoming the meeting as a positive opportunity, they said “the absence of an apology has been noted as a significant oversight in the pursuit of meaningful reconciliation and justice”.
They continued: “This morning the first minister said to Laura Kuenssberg that he offered the families an apology during this meeting. We can confirm that this simply never happened.”
In response, Swinney issued a statement saying he had expressed his “sincere sorrow and regret” during a “long, and at times emotional, call with families”.
He continued: “It was primarily an opportunity for them to tell me their experiences and for me to listen but I also tried to communicate to them how sorry I was about their experiences.
“I have obviously not communicated that effectively so I want to be crystal clear today. I am deeply sorry for the issues they have experienced at the QEUH and all that families have gone through.
“Their pain, trauma and suffering is unimaginable and I have said the conversation on Thursday was the start of a wider discussion to help families get the answers they deserve, beyond the work of the Scottish Hospitals Inquiry.”