James is said to be helping in the recruitment of all three players.
Allen, 39, is considering retirement, but sources close to the talks said he will join the Cavs if he returns for another season. After the Heat lost to the San Antonio Spurs in the NBA Finals, Allen told reporters he would like to play on the same team as James again.
Andersen, 36, is also being recruited by the Heat. One source said the Heat are determined to keep Andersen away from the Cavs. The talks between the Cavs and Andersen, however, were described as "not serious at this time."
Miller averaged 7.1 points and shot 46 percent on 3-pointers last season, playing in all 82 regular-season games.
Allen, considered one of the best perimeter shooters in league history, played 73 games and scored 9.6 points off the Heat bench last season. Andersen averaged 6.6 points, 5.3 rebounds and 1.5 blocks in 72 games.
James opted out of his contract with the Heat last month. After several weeks of mystery about where he would play next, he announced his decision to join the Cavs in a first-person account on Sports Illustrated's website Friday.