The former Rangers and Liverpool midfielder joined the Dens Park side on a two-year contract in September 2020 and captained them to promotion in his first season before finishing bottom of the Premiership last term.
The 36-year-old, who made 65 appearances for the club, scoring 11 times, said he took the decision to leave the Dark Blues following discussions with the board of directors about their plans for the future.
Adam has been appearing on BBC Scotland as a pundit recently but stopped short of confirming whether he plans to retire from playing.
In a statement posted on his Twitter account, Adam wrote: "It was always the dream to play for my home town club, and the last two years at Dundee FC have been really special for me and my family.
"To captain the club, and lea the team to promotion is something I'll always treasure and one of the highlights of my career.
"The supporters have been nothing short of incredible to me - I'll never forget how you've all treated me in the last two years.
"All that said, after talks with the directors about their future plans, I've decided to leave the club at the end of my contract this summer.
"I wish everyone at Dens Park all the best in leading Dundee back to the Premiership next season.
"As for my future, I'm looking forward to a new challenge which I'll consider over the next few weeks.”
Dundee also confirmed the departures of Liam Fontaine, Danny Mullen, Declan McDaid and Christie Elliot.