Stoudemire toured the Knicks' practice facility and met reporters wearing a Knicks cap, saying, "The Knicks are back," The New York Times reported.
Terms of the deal were not announced but a team official whose name was not reported told the Times it is a five-year maximum contract worth about $100 million. The official amount won't be known until the league's moratorium on free agent contracts ends Thursday.
The Knicks hope Stoudemire's signing may help them land LeBron James, Dwyane Wade, Chris Bosh or another big name free agent to the team, the report said.
The Times said one of the Knicks' other targets was Joe Johnson, but he elected to remain with Atlanta, agreeing to a contract Sunday.
By signing with the Knicks, Stoudemire is reunited with his former coach in Phoenix, Mike D'Antoni.