Mr Leeson, whose risk-taking deals brought the bank crashing down, met Peter Norris while making a show for BBC Radio 4.
He said: "Remorse is a word that is bandied around an awful lot, but I am sorry and my apology to Peter is genuine."
Mr Leeson, who was eventually sentenced to six and a half years in a Singaporean jail for his crimes, admitted he did not know whether his apology would "make any difference".
A Watford schoolboy-turned-City whizzkid, Mr Leeson's early career was a success and he was promoted within the bank. However things began to unravel when he started making losses and set up a secret account to hide them. He took more risks in a bid to recoup his losses, but things spiralled out of control and he fled leaving behind losses of 750 million and a note on his desk saying: "I'm sorry."
Speaking on Radio 4's The Reunion, he said: "I was losing tens of millions of pounds on a daily basis. Towards the end it could have been more - hundreds of millions. I was blinkered, attempting to extract myself from the situation.
"I never thought that the bank would collapse. I didn't even know what the bank was worth".
Mr Leeson eventually wrote a book about his experiences which became a film starring Scots actor Ewan McGregor.
• The Reunion will be transmitted at 11:15am on Sunday.