According to a WEEI source, the 38-year-old left-handed hitter was put through a workout that featured him doing drills at first base, a position he's never played in the big leagues.
Last season, Abreu played in 92 games for the Los Angeles Dodgers hitting for a .246 batting average, a .704 OPS, three home runs, 24 RBI and scored 29 runs.
In his time in the American League with the Anaheim Angels and New York Yankees, Abreu mainly served as a designated hitter and outfielder.
According to WEEI, the Red Sox have been exploring options for a backup at first base. Abreu could be a solid pick up if the money is right. Abreu has made $9 million a year for the last three seasons.