CHICAGO -- Jose Abreu has been racking up recognition, records and other accolades since he burst on to the scene with the White Sox this season. He won an award for the second time this season, but maintained the same cool and humble demeanor that he always does when asked about his accomplishments.

The 27-year-old White Sox first baseman was named the American League Rookie of the Month for June on Wednesday after hitting .313 (31-for-99) with six doubles, 10 home runs, 22 RBIs and 15 runs in 25 games. The 10 long balls were tops in baseball, and Abreu's homers, RBIs, .677 slugging percentage, 16 extra-base hits and 67 total bases in June led all rookies.

"In Cuba, we don't have that many awards. You never get that many awards," Abreu said through translator Lino Diaz. "That is probably why I look kind of cold when I get these awards here because we're not used to that."

If he keeps on his current pace, he'd better get used to it. Abreu also won the award in April -- becoming Chicago's first player to win the AL Rookie of the Month Award twice -- and won it in both months in which he did not miss significant playing time. Abreu missed the second half of May with an ankle injury, and the Astros' George Springer took home the honor.

Abreu, who is currently in second place to Detroit's Miguel Cabrera among AL first basemen in All-Star balloting, also received votes for the AL Player of the Month Award for June, although the Angels' Mike Trout won the award. He said he was named to six All-Star games in Cuba.

On the season, Abreu is batting .280 with an AL-best .627 slugging percentage, to go along with 26 home runs and 67 RBIs -- both tied with Baltimore's Nelson Cruz for the Major League lead entering play Wednesday night. Abreu has a 16-game hit streak going, a span in which he has batted .344 (22-for-64) with seven home runs and 16 RBIs.

He joins Jim Thome (2006) and Frank Thomas (1993) as the only players in White Sox history to hit 10-plus homers in a month twice in one season.

"I think some of the times you see the people that he is compared to, it doesn't matter who it is," said manager Robin Ventura, who sees Abreu as worthy of an All-Star selection. "Even the numbers he's putting up, whether the rookie part is in it at all, just the fact he's mentioned with the kind of people that he is, just shows how great a year he's had and what a force of a run producer that he is."

Abreu said once again he's not interested in participating in the 2014 Gillette Home Run Derby. He said he participated in five such contests in Cuba and never won because it's "difficult" to purposely try to hit home runs.

It hasn't proven difficult for him to hit home runs during live competition, and he's dedicated each one of those long balls to a person who helped him get to this point.

"Every time I hit a home run, I enjoy it, and more so because there's one person that really enjoys when I hit them, and that's my mother," Abreu said. "Every time I hit one, I know that she's happy."

The other players who received votes for the AL Rookie of the Month Award included the Rays' Kevin Kiermaier, the Yankees' Dellin Betances and Masahiro Tanaka, the Mariners' Roenis Elias, the Tigers' Nick Castellanos and the Royals' Yordano Ventura.