Dampier visited the Heat on Wednesday and was presented with a one-year offer for the veteran's minimum of $1.4 million after meeting with president Pat Riley and coach Erik Spoelstra, sources said.
Houston, which Dampier also visited this week, has offered Dampier a two-year, $4 million deal. The opportunity to win a championship alongside Miami's super trio of LeBron James, Dwyane Wade and Chris Bosh has Dampier favoring the Heat.
Though he averaged six points and 7.3 rebounds for Dallas last season, Dampier would be a big signing for the Heat. At 6-foot-11, 265 pounds, he would fill the only real hole on Miami's roster -- that of a big man who can physically match up with Orlando's Dwight Howard, Boston's Shaquille O'Neal and the Los Angeles Lakers' Andrew Bynum.
Chris Broussard covers the NBA for ESPN The Magazine.