Lee averaged 11.4 points per game for the Houston Rockets last season and hit 40 percent of his 3-point attempts.
The Celtics don't have the salary-cap space to sign Lee outright, but sources say the Rockets have let it be known that they're willing to participate in sign-and-trade deals with both Marcus Camby and Lee amid the increasing likelihood that neither of those coveted free agents will be returning to Houston.
Allen, who spent the past five seasons in Boston, agreed to sign with the reigning champion Heat on Friday, leaving the Celtics thin at shooting guard.
The Minnesota Timberwolves, Atlanta Hawks, Indiana Pacers, Dallas Mavericks and Phoenix Suns, sources said, are the other teams in pursuit of Lee. The Chicago Bulls have long been interested in Lee, as well, but the Bulls' limited financial flexibility means they'd also need a sign-and-trade deal to land the swingman.
ESPNBoston.com's Chris Forsberg contributed to this report.