Allen, one of Miami's top offseason targets, will also meet with the Los Angeles Clippers on Friday, the sources said.
NBA.com and The Associated Press had earlier reported Allen's planned visits.
Miami can offer Allen, who turns 37 this month and has spent the past five seasons with the Boston Celtics, only the mini mid-level exception, worth just over $3 million a year, while Boston is positioned to offer a two-year, $12 million contract, sources have told ESPN.com.
Celtics coach Doc Rivers talked to Allen on Sunday and called it a "stretch" that the NBA's all-time leading 3-point shooter will play elsewhere next season.
When asked about his status in a radio interview with "The Dan Patrick Show" on Tuesday, Rivers sounded much less confident.
"I don't know," Rivers said. "We're working hard. He has a lot of options and that's the bad news for us. It'll be interesting. I wish I had a feel on it. Usually I do. But I can tell you in this one I don't."
The Memphis Grizzlies have offered Allen their full mid-level exception, worth $5 million, for two years.
Like the Grizzlies, the Clippers can offer Allen the full mid-level exception. The Clippers view Allen as a starter, sources have told Broussard.
"I've talked to the Heat several times, and we'll continue to talk to them," agent Tony Dutt told the South Florida Sun-Sentinel on Monday. "We're just looking at all of the options at this point and we're going to sit down later in the week and figure out the top three or four spots that make sense and then he may visit those particular spots."
The Heat made just under 36 percent of their 3-point attempts this past season. Mike Miller (.453) and James Jones (.404) led the Heat in accuracy from beyond the arc, though Miller is sorting out what he will do next season as he deals with back and foot issues.
Allen would figure to be a perfect fit with Miami because the Heat want to surround LeBron James, Dwyane Wade and Chris Bosh with shooters who extend defenses and therefore create room around the basket for the Big Three to attack. That approach worked well for Miami in the playoffs -- the Heat were 9-1 when making at least eight 3-pointers in playoff games (7-6 otherwise), and they hit 14 in the Finals-clincher over Oklahoma City.
Allen has made at least 100 3-pointers in 15 of his 16 seasons, the exception being when he connected on 74 in the shortened 50-game schedule of 1998-99. He's established career bests for accuracy in each of the past two seasons, first making 44 percent of his 3's in 2010-11, then 45 percent this past year. His 2,718 career 3-pointers are the most in NBA history.
Information from ESPN The Magazine's Chris Broussard, ESPNBoston.com's Chris Forsberg and The Associated Press contributed to this report.