Marc Berman of the New York Post reported Rose is "expected to be interested" in bringing Anthony back to New York as a free agent.
Marc Stein of the New York Times reported earlier this week that Rose's official first day working for the Knicks will be on March 1.
Rose worked for CAA as a player agent with Anthony as one of his high-profile clients. He will be taking over the role previously held by Steve Mills, who left his position with the Knicks on Feb. 4.
Berman noted that Rose had been Anthony's agent since 2009 and helped "engineer his departure" from the Denver Nuggets. He was traded to the Knicks in February 2011 and spent seven seasons with the team.
Anthony spoke to Newsday's Al Iannazzone on Thursday night about the possibility of returning to New York with Rose running the Knicks.
"It's hard to say because I don't know what that situation is going to be,” he said. "The easiest thing to say is, 'His agent is there, he's coming back.' Until I sit down and see the whole plan, I don't know."
After a 12-month layoff, Anthony returned to the NBA in November with the Portland Trail Blazers. The 35-year-old is having a successful rebound season with 15.5 points and 6.4 rebounds per game. His 36.0 three-point percentage is his best since 2013-14 (40.2).
The Knicks (17-42) clinched a losing record for the seventh straight season after Thursday's 115-106 loss to the Philadelphia 76ers.