The 29-year-old joins the Dark Blues after leaving Heart of Midlothian following the expiry of his contract last month.
Forrest previously played for Livingston and Ayr United, where he emerged from the academy and became the club’s youngest goalscorer after scoring on his debut against Queen’s Park in the Challenge Cup.
The former Scotland Under-21 international chose Dundee despite interest from clubs in Scotland and England.
“As soon as Dundee were interested, I thought it was a good fit for me. The way they played last year, they were a match for any team in the league so I’m looking forward to getting started,” Forrest told Dundee media.
“Whether it’s helping off the ball or creating and scoring goals, I want to affect games as much as I can. Obviously, the competitive games have already started so I’m looking forward to getting into it straight away.”