The 27-year-old has agreed a two-and-a-half year contract at The Den.
The transfer is subject to the formalities being completed and Football League clearance.
Dons manager Karl Robinson admits that it will be hard to replace Williams, who was named both MK Dons Player of the Year and Players’ Player of the Year last season.
“(Shaun) is going to be a big loss to the Football Club. His contract was due to run out at the end of the season and although we offered him a new deal, he’s decided to move on and we’ve got to respect that," he told the club's official website.
“I’m just extremely grateful and thankful to Shaun for all he’s done for me and the Football Club and I wish him all the best in this new chapter of his career.
"He’ll always be welcome back at Stadium MK whether that be as a fan or as a player."