The last year of the deal is a team option, league sources told ESPN.com.
"I'm excited to be a part of another journey," Jack told ESPN.com via text message.
The Golden State Warriors renounced their rights to Jack on Friday as part of a deal to sign free agent Andre Iguodala. That cleared the Cavs to make a deal with the 29-year-old combo guard who is coming off a season in which he averaged 12.9 points and 5.6 assists for the Warriors.
Jack is expected to play a similar role in Cleveland as he did in Golden State, where he was part of a strong three-guard rotation with Stephen Curry and Klay Thompson. The Cavs have a young backcourt with All-Star Kyrie Irving and 2012 first-round draft pick Dion Waiters that Jack will now join.
Jack likely will be the Cavaliers' sixth man, backing up both guard positions, but he said his role with the Cavs has not been discussed.
"I have no clue," he said, when asked if he'll come off the bench.
Earlier this week, the Cavs reached an agreement with free-agent forward Earl Clark on a two-year deal with a team option for next season.
Information from ESPN The Magazine's Chris Broussard and ESPN.com's Brian Windhorst was used in this report.