Sources close to the situation told ESPN.com's Marc Stein on Tuesday the Brooklyn Nets were making a hard push to land Kirilenko. A source also told Stein the Golden State Warriors were among the bidders for Kirilenko.
The 31-year-old spent 10 seasons with the Jazz before joining CSKA Moscow in his native Russia.
Also on Wednesday, the Timberwolves agreed to a multi-team trade with the Phoenix Suns and New Orleans Hornets that will send center Robin Lopez and forward Hakim Warrick from Phoenix to New Orleans, while moving forward Wesley Johnson and a first-round draft pick from Minnesota to Phoenix, a person familiar with the deal told The Associated Press.
The deal also sends guard Jerome Dyson and retiring center Brad Miller's $5.1 million contract from New Orleans to Minnesota, allowing the Timberwolves to clear space under the NBA's salary cap as they pursue free agents. Minnesota also will receive two second-round draft picks from New Orleans, while the Hornets will receive cash.
Minnesota general manager David Kahn declined to confirm the deal when asked about it on a conference call Wednesday, saying only that he is "not allowed to comment on anything that hasn't been consummated."
The Timberwolves did announce the signing of 23-year-old Russian guard Alexey Shved on Wednesday.
Shved spent the last six seasons playing professionally in Russia and for his country's national team. Most recently, he scored a game-high 22 points to go with six assists to help Russia beat Nigeria for a spot in the London Olympics tournament. The Summer Games begin this weekend.
Though point guard Ricky Rubio is recovering from a torn anterior cruciate ligament in his left knee, the Wolves still have Luke Ridnour and J.J. Barea at point guard, plus Malcolm Lee and whoever else is acquired over the rest of yet another active offseason.
The 6-foot-5 Shved is a shooting guard first, but he can move to the point guard spot, too. Kahn, who first saw Shved play last year, was taken by Shved's creativity with the ball.
Information from ESPN.com's Marc Stein and The Associated Press was used in this report.