On Wednesday, Shams Charania and Jon Krawczynski of The Athletic reported Butler has provided the T-Wolves with a list of the "one to three teams" with whom he'd be willing to sign a contract extension.
However, he also reported Minnesota plans to keep Butler and try to make the playoffs this season.
As far as the Knicks' interest, team president Steve Mills said Monday at a fan event that "his team would not trade assets for a player it felt was attainable in free agency," per Wojnarowski.
While it could be considered a surprise that Butler left the Los Angeles Lakers off his list, Wojnarowski reported that LeBron James' arrival made landing with L.A. less appealing.
Minnesota acquired the 29-year-old Marquette product in June 2017 in a blockbuster deal with the Chicago Bulls. He averaged 22.2 points, 5.3 rebounds, 4.9 assists and 2.0 steals while shooting 47.4 percent from the field across 59 appearances last year.
Although Butler enjoyed on-court success, ranking 18th in the NBA in player efficiency rating, there have been rumblings about his unhappiness within the squad.
In July, Joe Cowley of the Chicago Sun-Times reported Butler was "frustrated with the nonchalant attitudes of younger teammates—specifically Karl-Anthony Towns—[and] does not intend to sign an extension with the Timberwolves."
Per Cowley, the four-time All-Star selection is also keen on linking up with Boston Celtics point guard Kyrie Irving next offseason—both players can decline player options in their contract to become free agents in 2019—to create a potential superteam in the Eastern Conference.
Krawczynski noted Timberwolves head coach Tom Thibodeau was "initially resistant" to the idea of a trade, but Butler remains hopeful of moving to a new team as the new season approaches.
Krawczynski also reported the request has "very little" to do with the reported issues with Towns and Andrew Wiggins and is more related to contract concerns.
Butler is scheduled to receive a $18.7 million base salary for the 2018-19 season, which ranks 12th among shooting guards, seven spots behind Wiggins, per Spotrac.
It's unclear whether there's enough time to complete a major trade before the campaign gets underway in less than a month. The Wolves are set to open the preseason against the Golden State Warriors on Sept. 29, and their regular season begins Oct. 17 when they take on the San Antonio Spurs.