Jiménez received four second-place votes out of 30 votes cast for the award, and also received eight third-place votes, according to data released by the Baseball Writers Association of America.
The votes were enough to earn him 20 points, leaving him just behind Tampa Bay Rays infielder Brandon Lowe in the voting.
Jiménez finished the year with a .267 batting average, slugging 31 home runs and driving in 79 RBIs for the White Sox. He appeared in 122 games for the South Siders.
Houston Astros outfielder Yordan Álvarez was the unanimous choice as the American League Rookie of the Year, earning all 30 first place votes. Baltimore’s John Means finished in second, followed by Lowe and Jiménez.
New York Mets infielder Pete Alonso won the National League Rookie of the Year award, earning 29 first place votes. Mike Soroka of the Atlanta Braves also received one first place vote and 25 second place votes.
The Manager of the Year and MVP Awards will be announced later in the week.