One evaluator compared Abreu to Andres Galarraga, the former big league first baseman.
In 42 games this year, Abreu batted .382 with 13 homers. He belted 35 home runs in 2012.
Marlins evaluators have been in the Dominican Republic all weekend, and Abreu has been doing private workouts for clubs.
There is certainly a growing market for Abreu, and the question is whether the Marlins can win the sweepstakes for his services, since he likely will command at least $50 million.
Abreu has not yet been granted free-agent status by Major League Baseball, so he isn't currently entertaining offers.
But Abreu isn't the only player the Marlins are scouting in the Dominican Republic. There is an MLB showcase of top international teens taking place. Many of these players won't be eligible to sign until July 2.
Miami has its eye on a couple of young pitchers from Cuba who are at the showcase.
Joe Frisaro is a reporter for MLB.com. He writes a blog, called The Fish Pond. Follow him on Twitter