"I was thrilled to be offered the job at Brighton and I was excited at the prospect of working with Sami Hyypia again - but, totally unexpectedly, I have been given an opportunity to work in the Premier League," Lee, who leaves Bolton Wanderers, told Brighton's website.
"I fully appreciate that this is not an ideal set of circumstances and I am very sorry for the inconvenience and any embarrassment my change of mind, after the announcement was made, has caused.
"However, at this stage of my coaching career, the opportunity to work again at the very top level of English football was not something I felt I could turn down."
Brighton chief executive Paul Barber said that it was a unique situation that he had never come across before.
"It is extremely disappointing for the club," Barber said. "Sami Hyypia and those of us who worked hard to bring him here, that Sammy Lee has now reversed his decision after agreeing to become assistant manager.
"Nobody at our club, or Bolton Wanderers, was aware of any interest from Southampton until several hours after we announced Sammy's appointment as our assistant manager. It's certainly not a situation any of us have come across before."
Both Hyypia and Koeman will manage in England for the first time next season.