Addison was operated on in San Francisco on Monday and Clough has received positive reports from America.
"We are told the operation has been successful," Clough told the club's official website. "Miles will stay out there for another week to allow the surgeons to monitor the early part of his recovery and then he will be returning home.
"We are delighted the procedure has gone well but neither Miles or ourselves are in any doubt there is a long road ahead before we see him back in a Derby County shirt. There is no specific time frame being put on his return.
"It is important everything is done properly and methodically. Miles is an important asset for this club and has a long career ahead of him. It would be foolish to jeopardise his long-term future by being impatient."