It is a one-year, a source told ESPNLosAngeles.com's Dave McMenamin.
Johnson's deal will be for the veteran minimum, the source said.
"Wesley is a young, athletic player who runs the floor well and is an excellent defender," general manager Mitch Kupchak said in a release. "He's a developing player who we think could become a good NBA player for years to come. He's also a great kid and we're happy to add him to our team."
The Los Angeles Times earlier reported the agreement.
Johnson averaged 8.0 points and 2.5 rebounds per game last year for the Phoenix Suns. He spent the previous two seasons with the Minnesota Timberwolves, who selected Johnson with the fourth overall pick in the 2010 draft.
Johnson averaged a career-low 19.1 minutes per game last season and started just 21 of the 50 games he appeared in for the Suns.