Stevenson will receive a one-year, $2.5 million contract, sources told ESPN The Magazine's Chris Broussard.
"He gives us a guy that has proven he likes to play defense and challenges guys defensively," King said. "Veteran presence that's played the game and understands it. So from that standpoint, he's won a championship, so he knows what they had to put in to win it."
Stevenson has played for five teams in 12 seasons, most recently for the champion Dallas Mavericks.
He averaged only 5.3 points per game during the regular season but came through in the clutch in the NBA Finals. With the Mavs trailing the Miami Heat two games to one in the series, Stevenson scored 11 points in Game 4 to help spark the Mavs to victory. Then, in the title-clinching Game 6, Stevenson hit three big 3-pointers with the Mavs trailing and Dirk Nowitzki in foul trouble to push the team to a lead.
Stevenson has career averages of 7.7 points and 2.2 rebounds per game. Although he started 54 games for the Nets last season, he'll likely come off the bench at shooting guard and small forward.
Deron Williams said that Stevenson was "definitely one of my suggestions."
"A player like DeShawn brings toughness and defense, defensive-minded," Williams said. "(He's) another shooter that can space the floor. I think he's going to be a good addition to our team, a guy that knows how to play basketball. He's been in the playoffs year after year. He's won a championship. He brings a winning attitude to this team, which is something we need."
Stevenson brings some help and ruggedness to the Nets bench. The 6-foot-5 swingman is often used to defend opposing teams' best player. He has a reputation for being a feisty player but will add some big-game experience to a young roster.
"We need guys with high basketball IQ's that are physically and mentally tough, that can guard multiple positions and knows how to play on the road," coach Avery Johnson said. "We weren't a very good road team last year so we need players that have had really good experiences with their previous teams of playing on the road and he had a lot of good road experience last year."
ESPNNewYork.com's Christopher Hunt contributed to this report.