The 33-year-old Anthony will waive his no-trade clause and his $8.1 trade kicker to accommodate the deal, according to ESPN.
A trade call is scheduled for Monday to finalize the deal with the league office, sources told The Vertical.
Anthony joins MVP Russell Westbrook and newly acquired All-Star forward Paul George in a reshaped and Western Conference contender. Both players reportedly played a significant role in recruiting Anthony to waive his no-trade clause for Oklahoma City.
The Thunder are making a major commitment with the Anthony deal, incurring $27 million in luxury tax for the 2017-18 season. He has two years and $54 million left on his contract, including a player option for $27.9 million on the 2018-19 season.
Talks between Thunder general manager Sam Presti and Knicks GM Scott Perry intensified in 24 hours before Saturday's agreement, league sources told ESPN.
Anthony could take part in the start of Thunder training camp on Tuesday in Oklahoma City.
The Knicks stated publicly on Friday that Anthony would return to the team for the start of training camp, but an increased level of urgency began over the weekend, The Vertical reported.
Anthony had a no-trade clause as part of the maximum contract he signed in 2014 and declined a trade from the Knicks throughout the turmoil of Phil Jackson's tenure as president of basketball operations.
Anthony averaged 22.4 points on 43.3 percent shooting along with 5.9 rebounds and 2.9 assists in 74 games with the Knicks last season, his seventh in New York and 14th in the NBA.