On the eve of their annual Media Day, the Miami Heat are still trying to shake up the roster.

According to two league sources, the Heat remain in pursuit of Minnesota Timberwolves swingman Jimmy Butler, who has been put on the trading block by Timberwolves owner Glen Taylor.

Butler has been unhappy in Minnesota for some time, according to sources, and made that clear to management after the team was eliminated from the playoffs in the spring. Butler then met with the team again early last week, and again expressed his desire to be traded.

His long-time coach, Tom Thibodeau, has been reluctant to move him, but according to an ESPN report Five Reasons Sports has confirmed, Taylor has stepped in and started dealing with other teams directly.

The Timberwolves made their choice clear on Saturday, when they signed Karl Anthony-Towns — with whom Butler is having personal issues — to a maximum five-year, $190 million extension.

As Miami Heat Beat and Five Reasons Sports initially reported Wednesday, not only have the Heat have been monitoring the Butler situation closely and trying to get involved, but Butler has made it clear to associates that he is intrigued by playing for them — even if the Heat were not on the “list of three teams” reported by other outlets.

Butler not only has a close relationship with Dwyane Wade, who just re-signed with the Heat, but is also intrigued by the Heat’s culture under coach Erik Spoelstra. Some of Butler’s issues with teammates have resulted from his feeling that they didn’t work as hard as he does.

We have learned that Spoelstra would be engaged directly in conversation with Butler, pending the Timberwolves’ permission, as he has been with other Heat targets, to explain how Butler would fit in the basketball plan. Spoelstra was the primary driver of the presentation to Gordon Hayward, which nearly worked even though Hayward had the option of playing for his former Butler University coach, Brad Stevens, in Boston.

There is a time element here. The Timberwolves have their Media Day on Tuesday, and while Butler has already been excused from participation, the team wants to focus more on the Towns extension and not the Butler circus.

The Heat’s Media Day on Monday is also an issue; it will be awkward if current players, especially those considered part of the core, are peppered with questions about their future. One issue: The Heat would need to review Butler’s medical history, and that could likely not be done until Monday.

The Heat have been reluctant to part with second-year center Bam Adebayo in any package this offseason. But Butler could change the equation on Adebayo or other young pieces, such as Josh Richardson and Justise Winslow.

He is 29, and has logged a lot of minutes, but he would project as the fourth- or fifth-best player in a depleted Eastern Conference. The former Marquette guard would also be an heir to his friend Wade, and a potential magnet for other free-agent targets.

Five Reasons Sports co-founder, Ethan Skolnick, and Miami Heat Beat writer and co-host, Leif Sylvander, contributed to this report.