Will they reward their 22-year-old star with a max-out contract extension this summer that will keep him in the fold for seven more years? Perhaps more important, will they pacify the natural power forward's reluctance to play out of position at center another season?
The early indicators, after a five-game loss to the San Antonio Spurs in the Western Conference finals: Yes and yes.
Bryan Colanagelo, Suns president and general manager, called locking up Stoudemire and guard Joe Johnson, a restricted free agent, a priority Thursday. Stoudemire will cost the Suns more than $100 million, and Johnson will cost much more than the $45 million he turned down last summer.
But the Suns appear intent on keeping together the core group that won an NBA-best 62 games and went from 53 losses the season before to the conference finals.
"We have an unlimited future," said Mike D'Antoni, the league's coach of the year. "This is one step."
"We far exceeded expectations," said MVP Steve Nash, the catalyst of a run-and-gun team that averaged 112 points in the postseason but fell short defensively. "Not a lot of people picked us to make the playoffs. To be in the final four is a terrific accomplishment, one I believe we can top next year. Hopefully, this is one of those building blocks to something better."
Stoudemire is the cornerstone, and he came out of his exit meeting with Suns coaches and officials smiling about his future role: point-center.
The Suns, in an apparent compromise to Stoudemire's reluctance to play center again next season in D'Antoni's small-ball lineup, sent him home for the summer as the team's designated go-to player next season.
He earned the promotion with a stunning postseason that included averaging 37 points against Tim Duncan and the Spurs and 29.9 points overall in the playoffs.
The redefined role will have him pick-and-roll more often with Nash on the perimeter and free him to shoot outside, drive inside or pass.
Did someone say Magic Johnson?
"It's playing the position I played this year, center, but with a different twist — more of a perimeter, inside-outside game," Stoudemire said. "I'll do a lot more."
He added: "I've always dreamed of being in that position, the go-to guy. With this team, once I polish my game up a little more, I'll be able to take us to the promised land."
The Suns, with the 21st and 57th picks in the draft, want to muscle up inside to take the heat off Stoudemire. They know they have a rare talent.
Said Spurs coach Gregg Popovich: "Amare's not a center. He's a very quick, very powerful, very explosive power forward who plays anywhere he wants."
D'Antoni said Stoudemire is "going to define how to play in the NBA the next 10 years. He's going to be the model that people will have to adapt to.
"He can shoot threes, lead the break, pass, dribble and rebound. His defense has to get better. But when he becomes that complete basketball player, we will win championships."
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