According to ESPN's Adrian Wojnarowski, Cavs owner Dan Gilbert was seen speaking with T-Wolves owner Glen Taylor at the NBA Board of Governors meeting Friday.
Per Joe Vardon of Cleveland.com, Kyrie Irving told Butler after the 2016-17 season to avoid joining the Cavs.
Irving was subsequently dealt to the Boston Celtics, and the Cavaliers now have a major vacancy when it comes to their go-to player since LeBron James signed with the Los Angeles Lakers in free agency.
Wojnarowski reported Sunday that Taylor wants to trade Butler within "the next several days" despite head coach Tom Thibodeau wanting to keep him.
Cleveland is in transition without James in the fold as it currently features a mix of experienced veterans and promising youngsters.
The Cavs' top player is All-Star big man Kevin Love, who Cleveland acquired from Minnesota in 2014 as part of a trade that sent Andrew Wiggins and other pieces to the Timberwolves.
Butler is a four-time All-Star who averaged 22.2 points, 5.3 rebounds, 4.9 assists and 2.0 steals per game last season.
After Minnesota acquired him from the Chicago Bulls for guards Kris Dunn and Zach LaVine and power forward Lauri Markkanen, he helped lead the T-Wolves to the playoffs for the first time since 2003-04.
Cleveland's most valuable trade piece is likely rookie guard Collin Sexton, who the Cavs selected with the No. 8 overall pick in the 2018 NBA draft.
If the Cavaliers are unwilling to part with him, however, they may struggle to put together an intriguing enough trade package.
Dealing for Butler would also be a major risk on Cleveland's part since he could leave in free agency next offseason.
While acquiring Butler wouldn't move Cleveland past the likes of the Celtics, Philadelphia 76ers and Toronto Raptors as the favorite in the Eastern Conference, it would make the Cavaliers a strong bet to reach the playoffs for the fifth consecutive season.