Anthony, who finished the 2013-14 NBA season second on the scoring list with an average of 27.4 points a game and was one of the few bright spots in New York's 37-45 campaign, could declare himself a free agent on July 1.
"I'd like to appeal to his better nature about winning," Jackson said. "That's the beginning of team play."
Jackson, the Hall of Fame coach who won a record 11 NBA titles in charge of the Chicago Bulls and Los Angeles Lakers, came out of retirement last month to run the Knicks, the team he broke in with as a player.
On Monday, he made a big move to put his stamp on the operation by firing head coach Mike Woodson and the entire coaching staff after the team failed to reach the play-offs.
Jackson, 68, promised a shake-up of the roster after the last game of the season but he made it clear he would like to rebuild with Anthony, who has said he would consider taking less than a maximum salary if it helped to assemble a winning team.
The new Knicks chief has his sales pitch all worked out.
"There's some obvious examples that are out there that I could point to," said Jackson, mentioning Tim Duncan of the San Antonio Spurs and the Miami Heat's Big Three of LeBron James, Dwyane Wade and Chris Bosh as players who took less money in order to surround themselves with better team-mates.
"A precedent has been set. The way things are structured now financially for teams, it's really hard to just have one or two top stars, or max players, and put together a team with enough talent.
"You've got to have people making sacrifices financially. So we hope Carmelo is true to his word ... and we'll present that to him at that time."
Seven-time All-Star Anthony, who turns 30 next month, could make nearly $130 million on a five-year contract from the Knicks if given a maximum salary. If he signed with another team, the maximum he could get is just under $96 million for four years.
As far as his coaching search, Jackson said he planned to talk with Steve Kerr, a TV broadcast analyst and former Phoenix Suns general manager, who won three NBA titles as a player under Jackson in Chicago.
Jackson said he approached Kerr last year when he thought he might be hiring a coach as head of basketball operations for a possible team in Seattle that did not pan out, and has kept in touch since.
"We had another conversation, a breakfast in January this year in which we talked about styles of coaching. We are in very similar space about coaching in a lot of ways. Philosophically we have a strong connection," said Jackson.
"Whether he's able to take a job like this, I don't know. I'll be having a conversation with him later on this month and talk to him about it and see where he's at about his desire to coach."
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