Sources said that the Portland Trail Blazers also expressed interest in Murphy but he is deciding between the two East powers.
In a flurry of deals right at last Thursday's deadline, the Celtics traded three big men and also opened up three roster spots. Part of the strategy, league executives said, was to make room for one or two free agents to be added after buyout season. Murphy is a top choice to fill one of those roles.
The Heat have struggled this season with rebounding and scoring from their bench and Murphy would fill a need there. Miami currently has 15 players on its roster and would have to release a player to make room for Murphy.
Murphy has not played since Jan. 7 and averaged just 3.6 points and 4.2 rebounds in 18 games for the Nets. But he has a history of being a solid outside shooter and rebounder. He averaged 14.6 points and 10.2 rebounds in more steady minutes last season with the Indiana Pacers.
News of Murphy's buyout was first reported by the San Jose Mercury News.
Miami also is interested in point guard Mike Bibby if he can obtain a buyout from the Washington Wizards. The Washington Post reported on Sunday that Bibby's agent, David Falk, will meet with the Wizards Monday to discuss buyout terms.
Bibby has one more season on his contract after this year, a factor that might make the buyout terms more complicated.
Brian Windhorst covers the NBA for ESPN.com.