Coutts, 30, a star of the Blades' promotion to the Championship two years ago, was the headline name on the club's list of departees as the club prepare for life in the Premier League next season.
The Scottish midfielder spent almost a year on the sidelines after suffering a horror leg break in a game at Burton in November 2017 and found it hard to re-establish himself at first-team level in 2018-19.
On the exit of Coutts, who was out of contract in the summer, Wilder said: "Sitting down with Paul was the hardest thing I have done in my career. "It is really unfortunate, his story. But he is on board. He understands.
"We are giving him the opportunity to find a new club. He is desperate to play.
"He wants to play 30 or 40 times a season and, unfortunately, we just cannot guarantee him that."
Also leaving Bramall Lane when his deal expires at the end of June is versatile defender Martin Cranie, 32, who made a key contribution in the run-in.
Fellow out-of-contract players Conor Washington, Daniel Lafferty and Caolan Lavery will also leave.
Contracted players Leon Clarke, Jake Wright, Ched Evans, Samir Carruthers, Nathan Thomas, Ben Heneghan and Ricky Holmes have been placed on the transfer list.
Loanees Dean Henderson, Kieran Dowell, Scott Hogan, Gary Madine and Marvin Johnson all return to their parent clubs, with Henderson strongly linked with a permanent move to the Blades this summer.