According to ESPN's Adrian Wojnarowski, the T-Wolves are seeking quality veterans, top prospects, future assets and cap relief in any deal involving Butler. He added that teams are "skeptical" if head coach and president of basketball operations Tom Thibodeau truly wants to move Butler.
Wojnarowski and Zach Lowe later reported that interested teams believe Thibodeau is making counter offers that he "knows teams will never accept," adding that "Minnesota's counter was a non-starter for Heat president Pat Riley."
The Heat "had been willing" to take on Gorgui Dieng as part of a Butler deal, per Wojnarowski and Lowe.
Sam Amick of The Athletic later reported the Sacramento Kings "remain interested in serving as a third team, with a focus on landing a 2019 first rounder." However, Amick added that "as of late last night, no scenarios were even remotely firm."
Lowe noted that while the situation could "change on a dime," things are currently "volatile" in Minnesota.
Marc Stein of the New York Times reported Wednesday that Butler's preferred destination should he get traded is Miami.
Wojnarowski was the first to report last week that Butler had requested a trade, though he listed the Los Angeles Clippers as his preferred landing spot along with the New York Knicks and Brooklyn Nets as possibilities.
Thibodeau and Butler have history together dating back to Thibodeau's time as Butler's head coach with the Chicago Bulls. He traded for Butler last offseason in a deal that sent guards Kris Dunn and Zach LaVine and power forward Lauri Markkanen to Chicago for Butler and center Justin Patton.
Per Wojnarowski, Thibodeau objected to the idea of trading Butler last week, but owner Glen Taylor is motivated to get a deal done.
Last season, the 29-year-old Butler averaged 22.2 points, 5.3 rebounds, 4.9 assists and 2.0 steals per game. He also led Minnesota to its first playoff berth since 2003-04.
Despite the success, Wojnarowski noted that Butler doesn't have a good relationship with T-Wolves center Karl-Anthony Towns, who recently signed a max contract extension.
Butler would have the opportunity to be the go-to guy on a team like the Heat, and Miami likely has the assets needed to acquire Butler. Since Butler can opt out of his contract at the conclusion of the 2018-19 season, however, the Timberwolves may have to lower their asking price.