The England midfielder, who has spent his entire career with Rovers, agreed to join Birmingham on Wednesday and will sign for the Midlands club on completion of a medical.
"It was probably inevitable that David Dunn would move on after he refused the offer of a new contract," Williams told Rovers' official website."We all wanted him to stay, but these things happen.
"He was a local lad who had come through the system and he did well for the club. But football is a fluid business and we all have to move forward."