The skybet Championship side had earlier criticised Celtic over discussions on the move and revealed they were considering legal action.
Midfielder Shaw has followed up the Hillsborough statement with his side of the story in a social media post where he revealed the lure of “one of the most famous and prestigious clubs in the world” was too good to turn down – but he would give his all to his hometown club until the time for his move north.
“After discussing the options with my family, I have decided to take on a new challenge and so, I’ve agreed to join Celtic, one of the most famous and prestigious clubs in the world, on July 1.
"Until the last kick of the ball of the final game of this season, I will continue to give my all for Sheffield Wednesday and to help my team-mates.
"I work hard every day in order to make my family proud and I’ll always give everything in training and in games. I’m determined to contribute as much as possible to the team, so we have a successful end to the season.
"Sheffield Wednesday is a massive and brilliant club and I’m extremely grateful for everything the club has done for me. The club will always be special for me as I grew up a fan and nothing can change that for me.
"In the end, I based my decision purely on the football aspects. By joining Celtic next season I will fight for a place in a team challenging for titles, trophies and to play in Europe which is something I’ve always dreamed of.
"I am determined to play at the highest level I can possibly achieve in my career. I thank Celtic for this amazing opportunity, which is impossible for me to turn down.”
A deal for defensive midfielder Shaw, who scored his first senior goal in the last game of 2020, was revealed by former Celtic boss Neil Lennon a month ago.