According to Brian Windhorst and multiple league sources, there is "mutual interest between Allen and the Heat":
The most the Heat can offer is a contract starting at $3 million per year, which is known as the "taxpayer's mid-level." Allen could get more elsewhere, but the Heat do offer an attractive portfolio that goes deeper than cash.
Of course Miami is interested in Ray; there were rumors that their best perimeter shooters, Mike Miller and James Jones, are both considering retirement after winning their chip and Ray Allen is, well, the greatest catch-and-shoot wingman of all time. And sure, I don't doubt that Ray is interested in the Heat. They just won the championship and South Beach does not have state income tax or snow in February. But here's the thing: none of the dominoes have fallen for Boston. Garnett has yet to reveal his decision, the draft is this week, and Danny hasn't filled out the roster with returning free agents (like Ray himself) or younger players to ease Ray's return from bone spur surgery.
Everyone's penciled in Avery Bradley as next season's starting shooting guard but I don't think that's set in stone just yet. Even if Ray's pride took a hit when he became the team's sixth man at the start of the playoffs, he could win that job back. Regardless, I don't think it's an issue for Ray. People look at his career and notice that he's played for three teams in his career, but I think he's found a home in Boston. My bet is that if/when KG re-ups, Ray follows and retires a Celtic. I'll even go as far to say that if he plays two more years, #20 goes up in the rafters. Does Milwaukee or Seattle (or Miami) retire his number? Doubtful. Ray doesn't seem like a ring chaser to me. I'm sure winning another championship is important to him, but he can do that right here and not change zip codes. Ray's legacy will always be tied to the Big Three. A ring in Miami would never really belong to him. He'd just be along for the ride. Now, taking out Miami next season in a rematch and taking another trip to The Finals with Pierce and KG again, that's special.