The All-Star forward will receive a five-year, $99.7 million contract that will reunite him with coach Mike D'Antoni.
Stoudemire's agent, Happy Walters, said last week there was a "structure of an offer on the table," but no actual offer was put forth by the Knicks until they met with Stoudemire at Madison Square Garden on Monday afternoon. The deal cannot be signed until Thursday.
"The fact that Amare really wanted to come here and stepped up front, it got to the point that we had to acknowledge that," team president Donnie Walsh said. "That means something to us."
The Knicks believe they are still in the hunt for James (and, to a lesser degree, for Dwyane Wade), but Sunday night's news that James planned to delay making a decision until Wednesday or Thursday helped force their hand in regards to Stoudemire.
New York has been assured the Stoudemire agreement will not be an impediment to their chances of signing James or Wade, sources told ESPN.com, and the Knicks ultimately decided it might give them an advantage in the LeBron sweepstakes because James wanted the Cavs to acquire Stoudemire last February at the trading deadline.
"I feel great, really, about being a pioneer," said Stoudemire, who said he had spoken with James several times recently. "I think he feels great about it. It's a situation where no one wanted to make that first move, and I felt confident enough to take that first step, and hopefully now we can bring in a few guys to join us."
"I'm definitely going to reach out to him again and see if we can get him to come. I really have no reading on which way he is leaning at all, but again, he's totally open right now, he hasn't made a decision yet, and hopefully I can get him to choose us," Stoudemire said.
The agreement effectively removed the Knicks from the pursuit of Chris Bosh, and it significantly reduced the chances of David Lee returning to New York. The Knicks could still use Lee in a sign-and-trade deal if they miss out on James and Wade, but Lee could choose to take a five-year offer from another team between now and Thursday.
And between now and then, New York's management will remain cautiously optimistic that James will choose the Knicks, who spoke with James last Thursday, and with Wade and Bosh on Friday before having a second meeting with James' representatives on Saturday.
"I'm not sure where he's leaning. If he's leaning more toward New York, that would be a great start for us," Stoudemire said.
James has also received recruiting pitches from the Cavaliers, Bulls, Nets, Clippers and Heat.
"If somebody wants a real good basketball player to play with and win a title ... this could help get the dominoes falling," D'Antoni said.
Stoudemire arrived in New York on Saturday, head breakfast with D'Antoni on Sunday and met with team officials, including owner Jim Dolan, on Monday afternoon.
Stoudemire had spent his entire career in Phoenix, where he played under D'Antoni before the coach came to New York in 2008. Stoudemire averaged more than 20 points in every season they were together and immediately becomes the best player D'Antoni has coached since leaving the Suns after the 2007-08 season.
Stoudemire's days with the Suns ended late last week when the team agreed to $48 million worth of deals with forwards Hakim Warrick and Channing Frye. The sides had discussed an extension, but the Suns looked elsewhere after they'd reached a stalemate.
Stoudemire said he understood owner Robert Sarver's position and wasn't disrespected by the Suns refusal to give him a max deal that would have paid him millions more. He said he's always loved New York and wanted to play here since the Knicks passed on him in the 2002 draft.
Signs throughout the entrances to Madison Square Garden showed the player pictured in a Knicks uniform, wearing the No. 1, and marquee on the Seventh Avenue side of the building read "Welcome, Amare Stoudemire."
The Knicks desperately needed to come away with something in free agency after spending most of the last two years clearing enough salary-cap space to sign two maximum-contract players.
Chris Sheridan covers the NBA for ESPN.com. Information from ESPN.com's Chris Broussard and The Associated Press was used in this report.