Walker has passed a Milan medical and will join on-loan to the end of the season with an option to buy.
"A huge thank you to so many people, the coaching staff, the kit men and all the backroom staff who work tirelessly behind the scenes," he wrote.
"You make every day enjoyable and provide the platform for us to perform at our best.
"To my team-mates, from the moment I walked through the door I felt at home. Thank you for the great memories and for all the success we have shared together. You are friends, but also family for life.
"To Pep Guardiola, thank you for believing in me and working so hard to bring me here in 2017. Together, we've celebrated 17 trophies, and your guidance has helped shape me into the player I am today. I'll be forever grateful."