The potential deal comes amid reports that the Timberwolves are interested in signing the 31-year-old Kirilenko.
In the deal, the Phoenix Suns would send Robin Lopez and Hakim Warrick to New Orleans, and the Suns would get Wesley Johnson and a future first-round pick from Minnesota. The source said New Orleans would trade Brad Miller to the Suns. It's unclear what Minnesota would get from the deal.
New Orleans and Phoenix discussed a similar four-team trade last week, with Boston and Houston replacing Minnesota, but the Suns pulled out as the deal fell apart. That's when the Timberwolves entered the picture.
Earlier Tuesday, ESPN's Marc Stein reported the Brooklyn Nets were pushing hard to bring Kirilenko back to the NBA, despite their own limited salary-cap flexibility. Kirilenko, who spent last season with CSKA Moscow, wants to return to the NBA. Sources said that Team USA guard Deron Williams, Kirilenko's former Utah Jazz teammate, was part of the effort to recruit Kirilenko.
Sport Express in Russia on Tuesday quoted Kirilenko saying: "I want to decide my new team before the Olympics. There are several options in the NBA, but the final choice has not been made yet."
Kirilenko also told Ria Novosti, another media outlet: "Which club in particular, we'll know in a day or two."
Information from ESPN.com's Marc Stein was used in this report.