According to the newspaper, the Suns also will receive a conditional first-round pick in the deal.
Thomas, 32, averaged 11.5 points and 10.4 rebounds for the Knicks last season and was the team's best interior defender, covering both power forwards and centers.
Suns general manager Bryan Colangelo had said his offseason priority was to add a defensive low-post presence after his team lost in the Western Conference finals to the San Antonio Spurs, 4-1.
The 6-9 Thomas will be paid $14 million over the next two seasons and has a player option for $8.1 million in 2007-08.
The move will enable the Suns to move power forward Shawn Marion to small forward. They also plan to re-sign shooting guard Joe Johnson, who is eligible to become an unrestricted free agent in July, and sign Stoudemire to a contract extension.
Richardson, 25, signed as a free agent with the Suns last season and has five years and $37.7 million left on his contract. After helping the Suns to the league's best record at 62-20, Richardson was stunned by the reported trade.
"It definitely is a shock, but I also understand it's a business decision," Richardson told the Arizona Republic. "It's something that comes along with being a NBA player. I've gotta roll with it."
The 6-6 Richardson set a club record with a league-leading 226 3-pointers last season and averaged 14.9 points. In the playoffs, however, Richardson slipped to 11.9 points and was held scoreless in the series-clinching loss to the Spurs.
In New York, Richardson likely will start at small forward ahead of Tim Thomas, who is being shopped by Knicks general manager Isiah Thomas.