Though still a member of the Thunder, Anthony and the team are reportedly working together on parting ways, whether through trade, a buyout or waiving via the stretch provision.
Oklahoma City needs Anthony to waive his no-trade clause to allow him to leave.
If he chooses to sign with Houston, the Rockets can offer him their mid-level exception, which amounts to $5.3 million.
The moves add to uncertainty for the Rockets around Swiss free agent center Clint Capela, who has not agreed to a new deal.
Heat coach Erik Spoelstra advocates signing Anthony, according to the ESPN report.
The 10-time All-Star averaged 16.2 points on 40.4-percent shooting.
But Anthony has averaged 24.1 points and shot 44.9 percent in 15 National Basketball Association seasons and has been one of the NBA's premier scorers of the past 20 years.
Worryingly, his presence on the court saw the Thunder's team plus/minus rating slide to negative 14.3 during the playoffs. The 11-year National Basketball Association journeyman has also played for the Celtics, Timberwolves, Mavericks, Nets, Pacers, Suns and Heat. The 31-year-old averaged 7.5 points, three rebounds, and 1.2 steals in 25.6 minutes per game across 61 games (15 starts) in Houston.