While sources told ESPN.com's Brian Windhorst the Heat have made Allen their top offseason priority, sources confirmed to ESPN The Magazine's Chris Broussard the Boston Celtics and Memphis Grizzlies are also heavily pursuing Allen.
Sources said the Celtics, who Allen has spent the past five seasons with, are prepared to double the yearly salary the Heat can offer Allen, while the Grizzlies have offered Allen their full mid-level exception, worth $5 million, for two years.
Miami can only offer Allen its mini-mid-level exception worth $3 million. The Celtics, who have cap space, Allen's Bird rights, and who have made re-signing Allen their top priority now that Kevin Garnett is back in the fold, are ready to give Allen a two-year, $12 million contract, according to sources.
Atlanta, Dallas and the Los Angeles Clippers have also expressed interest in the 36-year-old sharpshooter.
Allen will take his time and go through the process, considering the financial aspects of each offer, his role on the team, and the team's ability to compete for a championship, sources told Broussard.
Celtics coach Doc Rivers talked to Allen on Sunday and called it a "stretch" to see Allen play elsewhere next season.
"We really want him back and I really think at the end of the day that he will be back because this is such a good fit for him. We need him back and we're trying to get him back," Rivers said after completing a morning round of golf in greater Boston. "We can offer him more money. Yes, there are a couple of teams way under the cap who might be able to offer more money. But other than that, we can offer him more (than Miami or Memphis). He already lives here. He knows his role and everything else. It's a stretch for me to see him go somewhere else."
Rivers said Allen gave no hints as to when he'll decide, but the coach was hopeful Garnett's decision to return to Boston will help convince Allen.
Free agency officially opened on Sunday at 12:01 a.m. ET, but players cannot officially sign until July 11.
Information from ESPN The Magazine's Chris Broussard, ESPN.com's Brian Windhorst and ESPNBoston.com's Peter May contributed to this report.