According to a league source, the Marlins have serious interest, and they plan to present a substantial offer for the 26-year-old.
Abreu was impressive in front of the scouts and evaluators, demonstrating his power in batting practice and in two scrimmages. In his first at-bat, he belted an opposite-field home run, and he showed his ability to hit the ball to all fields.
Abreu worked out on Monday and Tuesday at the Yankees' complex in Boca Chica. The Marlins' baseball school in the Dominican Republic is located only minutes away.
In attendance for the Marlins were general manager Dan Jennings, international operations director Albert Gonzalez, Double-A Jacksonville manager Andy Barkett and pro scout Dave Roberts.
The Marlins certainly aren't alone in their interest.
In the National League East, the Mets also had a large representation.
The Giants, White Sox, D-backs and Rangers also are said to have serious interest.
As a recruiting tool, the Marlins are hopeful Abreu will be attracted to Miami's large Cuban community. Marlins Park is located in the Little Havana section of Miami.
And the team has two Cuban natives on its roster, All-Star pitcher Jose Fernandez and shortstop Adeiny Hechavarria.
Upgrading their offense is the Marlins' top off-season priority. The club finished last in the Majors in most significant statistical categories, including runs (513), home runs (95) and batting average (.231).
Abreu draws comparisons to former big league slugger Andres Galarraga, a big presence with plenty of power and not a lot of mobility.
Although Abreu profiles more as a first baseman, he took some grounders at third base. At that position, he moved around similar to Detroit's Miguel Cabrera.
Projections are Abreu is not a plus defender at third.
His bat is what the Marlins are coveting, because they feel he could offer legitimate protection in the order to Giancarlo Stanton.
Abreu is represented by Miami-base agents, Barry Praver and Scott Shapiro.
The bidding on Abreu is expected to take place in a couple of weeks.
Marlins first baseman Logan Morrison is arbitration eligible for the first time. If the team does sign Abreu, it raises more questions for the front office. The team could use Abreu at third base or perhaps weigh trading Morrison.