Former Heat forward Mike Miller, who like James joined the Cavs on a two-year deal, said last week he was making a recruiting effort to lure Allen to the Cavaliers, pointing to the veteran as the kind of player that would help Cleveland in key spots in the playoffs.
Allen, who will turn 39 on July 20, played 73 games for the Heat in the regular season and all 20 postseason games last season.
He averaged 9.3 points per game on 41 percent shooting in the playoffs.
Allen has talked with multiple teams to investigate his options, sources have told ESPN.com's Brian Windhorst.
Ray Allen Ready To Join LeBron In Cleveland?
Brian Windhorst discusses the report that Ray Allen is leaning toward signing with the Cavaliers.