Of course, we need the general disclaimer that Kyrie's preferred destinations matter little, as he'll have to go wherever they decide to trade him. He's also signed for the next two seasons, so pretty much every team in the league would be willing to pay full price in terms of trade value for two guaranteed Kyrie seasons. Others who have suggested otherwise are completely misguided in my opinion, but the Paul George trade shows all it takes is one dumb GM to value players entirely incorrectly.
The Timberwolves could in theory put together a good offer, if they were willing to make Andrew Wiggins available, and it's at least possible Cleveland is still very high on the guy they drafted as their future cornerstone just three years ago, weeks before LeBron came home and changed everything.
Let's look at two trades that could net all teams involved something they're happy with.
Cleveland receives F/G Andrew Wiggins, F/C Gorgui Dieng, C Cole Aldrich, the 2018 OKC First Rounder, and the Wolves 2020 first round pick, top-1 protected
Minnesota receives PG Kyrie Irving, G Iman Shumpert, Future 2nd rounder
With LeBron playing basically every position, he and Wiggins could easily start together despite both being natural small forwards. Gorgui has become a very solid player and would be a tough loss, but likely would need to be included to put this deal over the top. Aldrich for Shumpert gives the Cavs an expiring contract while the Wolves get a useful wing player they badly need for depth purposes.
Now, if Cleveland isn't all that interested in Wiggins, maybe due to his upcoming max contract extension, let's take a look at the next trade, which is a 3-team deal.
Minnesota receives PG Kyrie Irving, G Iman Shumpert, F TJ Warren
Phoenix receives SF Andrew Wiggins, PG Tyus Jones, C Cole Aldrich, 2018 OKC Pick, Future Cavs 2nd rounder
Cleveland receives PG Eric Bledsoe, F/C Gorgui Dieng, 2020 Wolves 1st round pick top 1 protected
The Suns would have an enticing core with this kind of a trade, as a future lineup of Booker-Josh Jackson-Wiggins-Chriss-Bender for stretches of games would create matchup nightmares all over the court. The Wolves get two key bench players and Kyrie, but give up almost every asset they have left to do so. The Bledsoe-LeBron connection is well documented, and Gorgei's ability to play the 4 at times would allow the Cavs to explore Kevin Love trades if they still want to move on from him somehow.
In this scenario, look at the Wolves lineup:
PG - Kyrie/Teague/Butler or Vet Min FA?
SG - Butler/Crawford/Shumpert
SF - Warren/Shumpert/Vet Min FA
PF - Gibson/Bjelica/Vet min Fa or Patton
C - KAT/Vet Min FA/Patton
They'd need to sign 3-5 players to fill out their roster, and with a Kyrie-Butler-Kat core they'd likely get a few solid veterans willing to jump on for veteran minimum deals. Man, how fun would that team be. Let me dream for a few days before you trade him elsewhere please, Cleveland.