Tucker was the top defender and acknowledged leader in the Suns' surprising turnaround last season. He was a restricted free agent and several teams had shown an interest in him.
Tucker's agreement was first reported by azcentral sports, which includes the Arizona Republic.
The person also confirmed that restricted free agent Isaiah Thomas visited the Suns on Thursday with more meetings set for Friday.
The Suns envision Thomas as a third point guard behind the double-point guard starting duo of Goran Dragic and Eric Bledsoe.
Adding Thomas would create a crowd at point guard. The Suns also have Ish Smith, second-year pro Archie Goodwin and first-round draft pick Tyler Ennis at the position.
Thomas averaged 20.3 points and 6.3 assists last season with the Sacramento Kings, who are not expected to match any significant offer sheet he signs.
Phoenix, with a load of cap space and a talented young roster, made a long-shot run at LeBron James before turning its attention elsewhere.
The person emphasized that the talks with Thomas are unrelated to Bledsoe's situation.
Bledsoe is a restricted free agent but general manager Ryan McDonough has said the Suns would match any offer sheet he signs. The Suns are attempting to reach an agreement with Bledsoe before he inks any offer. Bledsoe's agent, Rich Paul, also represents James, and could be occupied by those negotiations.
Tucker was the heart of the energized Suns team that went 48-34 and just missed the playoffs in the tough Western Conference. Tucker, who played at Texas and helped recruit Kevin Durant to the Longhorns, was with Toronto as a rookie in 2006-07, then played in Israel, Ukraine, Greece, Italy and Germany before coming to the Suns in 2012.
The 6-foot-5 small forward started all but one game, averaging a career-high 9.4 points per game. He shot 39 percent from 3-point range, almost always from the corner.
Earlier, Phoenix lost forward Channing Frye, who agreed to a contract with the Orlando Magic.