Regan was heavily criticised for his handling of the Ibrox club's demise which ended with Ally McCoist's side being denied entry into the Scottish Premier League and voted into Irn-Bru Division Three.
The SFA supremo had to be given safety advice after being sent intimidating messages online and through the post.
Regan told a number of newspapers: "When you start to get threats on things like Twitter and emails and letters through the post that are uncomfortable, let's say, then you have to listen seriously to what the police are saying.
"We have had visits from the counter-terrorism unit and I've had personal discussions with the police from the perspective of my own safety."
Regan, who has dropped his Twitter account, has kept his family in England and lives alone in Scotland, where he had to contend with a security clampdown, but insists he never once thought of quitting his role.
He said: "I've had the conversations with police about people outside, police cars driving past outside my flat and so on.
"But it's never once made me think about walking from the job. I remain committed to the job and will continue to give 100% to implementing the changes.
"My family spend time down south and I live here on my own. My work life and issues that go with it are mine and I keep the family out of it."
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