A source close to Anthony told ESPNNewYork.com that an announcement of a Knicks deal could come from the seven-time All-Star as soon as Sunday.
The contract will be for more than $122 million but less than the five-year, $129 million max figure, a source told ESPN The Magazine's Chris Broussard.
Knicks president Phil Jackson indicated Saturday night to reporters that he was taking a patient approach.
"We don't have anything yet to stack up so rather than talk about it, I won't," Jackson said. "We'll wait. We'll be there."
Anthony has said that he'd be open to accepting a contract that was worth less than the maximum allowed. Doing so wouldn't give the Knicks any extra money to offer free agents this summer, as they are over the salary cap. But if Anthony took a pay cut, it could provide more money for the Knicks to offer free agents in the summer of 2015, when they are expected to have at least $20 million available.
"It's exciting to hear that it's still possible and that at least it's close to being real," Knicks coach Derek Fisher said. "But I've been around a long time, so until it's done and it's real, we can't assume that it's a definite thing."
Fisher said he hasn't spoken to Anthony since Saturday's reports about the two sides closing in on a deal surfaced. Now, he's waiting to hear from the men up top.
"Phil [Jackson] and Steve [Mills] will know more than me in terms of the timing and if it's going to happen really soon," Fisher said of the team's president and general manger, respectively. "But I'm definitely encouraged by what I'm hearing. It sounds a lot more positive."
The Bulls had moved Anthony down on their priority list as recently as Friday, as sources told ESPN.com's Marc Stein that they were prioritizing their pursuit of free-agent forward Pau Gasol even ahead of their long-running efforts to sign Anthony, fearing that Anthony was likely to stay with the Knicks. Gasol decided to sign with the Bulls on Saturday, he tweeted.
Information from ESPNNewYork.com's Ian Begley and ESPNNewYork.com contributor Fred Katz and was used in this report.