According to the paper, sources indicated the Cavs and Suns have been discussing a deal involving the five-time All-Star for nearly a month.
The two teams have collaborated on deals before, including a trade that sent Shaquille O'Neal to Cleveland in the summer of 2009.
In recent weeks, Stoudemire has heard his name pop up in one trade rumor after another. Suns coach Alvin Gentry had a discussion with his big man before the Suns' victory over the Nuggets last week about all the speculation, reminding Stoudemire the situation is completely out of his control.
"I told him he's got to try to put it behind him and move on because whatever happens is going to happen," Gentry said after the game. "It always happens this time of year, there's someone that's always the talk -- it just happens to be him this year."
What's fueling the trade rumors is the fact that Stoudemire can opt out of the final year of his deal after the season. The Suns could decide to trade Stoudemire to get some value from him rather than let him walk away.
"It hasn't been explained to me why every year I'm the subject of trade talks. At the same time, it's a business," Stoudemire said of the situation. "They've never really explained to me why they want to trade me. At the same time, I can't control what I can't control. I stay out of it, and try to keep my focus on the court."
The NBA trade deadline is Feb. 18.
Stoudemire is averaging 21.2 points and 8.6 rebounds on the season for the 31-22 Suns.
Information from the Associated Press was used in this report