O'Neal's faves: Mavs, Nuggets, Celtics
"Those are my favorites," O'Neal told ESPN.com "Those are teams that are going to compete [for a championship] next year."
Later in the day, his decision might have become more clear, with a source close to the situation telling ESPNDallas.com that O'Neal will likely sign with the Celtics.
All three teams are believed to be offering a portion of the mid-level exception to sign him, and O'Neal called the decision perhaps "the most important of my career because I know this is kind of my last run at it" in terms of title-chasing.
O'Neal averaged 13.6 points, 6.9 rebounds and 28.6 minutes in 70 games last season for the Heat, who are expected to renounce their rights to the former All-Star as part of the cap space-clearing exercise required to create sufficient cap room for the incoming Chris Bosh alongside a re-signed Dwyane Wade and Miami's other free-agent targets.
All three O'Neal suitors are searching for frontcourt depth.
Boston is expected to start the season without the injured Kendrick Perkins; Denver's Kenyon Martin and Chris Andersen are likewise recovering from injuries; and Dallas faces a potential void at center amid uncertainty about its ability to re-sign Brendan Haywood and with the Mavericks actively shopping Erick Dampier's non-guaranteed $13 million contract for an impact player.
O'Neal, 31, is an unrestricted free agent for the first time in his career and had his best years in Indiana under Mavericks coach Rick Carlisle before persistent knee trouble derailed him.
Marc Stein is a senior NBA writer for ESPN.com. ESPNDallas.com's Jeff Caplan contributed to this report.