Anthony met with the Lakers last Thursday and was said to be "agonizing" while determining whether to return to the Knicks or join the Lakers or Chicago Bulls.
"He will have something for everybody on Thursday," said a close friend of Anthony, according to the newspaper. "He is really torn because this is the biggest decision of his career. But he wants to get it done in New York. He told me he believes in Phil."
That would be former Lakers Coach Phil Jackson, who became the Knicks' president earlier this year and reportedly formed a close connection with Anthony.
If Anthony chooses the Knicks, the Lakers' chances of a reunion with Pau Gasol drop drastically.
Gasol, 34, wants to return to the place he has called home since 2007 but wants a solid roster around him, which the Lakers are currently not able to provide with a "nucleus" of the aging Kobe Bryant and injury-prone Steve Nash.
Gasol has been courted by San Antonio, Oklahoma City, Miami and Chicago. He made $19.3 million last season but was looking at a substantial pay cut.
In a free-agent system that often incentivizes big-money players to stay with their teams, Anthony, 30, would have had to accept a four-year, $95.9-million contract with the Lakers instead of a five-year, $129.1-million deal with the Knicks.
Anthony, who was born in Brooklyn and played at Syracuse, averaged 27.4 points for the Knicks last season, along with career-highs in rebounds (8.1) and minutes (38.7).
The Lakers have met with the agent for LeBron James but appear to have only a small chance of signing James to a free-agent contract.