By Bruce Levine–

GLENDALE, Ariz. (CBS) — A relentless pursuit of left-handed power has been the objective of the Chicago White Sox front office since last October.

The team’s brass has done a quality job of filling gaps at three infield positions and behind the plate. The energy and talent brought in by the additions of Todd Frazier, Brett Lawrie, Jimmy Rollins, Dioner Navarro and Alex Avila appear to be significant upgrades across the board.

Despite this improvement, the White Sox know that a big bat from the left side will be needed to have proper run production balance. Presently, this plan may be incomplete.

White Sox owner Jerry Reinsdorf has always allowed his front office to add important players despite having to go over budget from time to time.

Chicago has burnt the midnight oil talking to clubs about potential trades. The White Sox missed out on free-agent target Alex Gordon, who decided to return to the Royals on a long-term deal. They were right there too on Yoenis Cespedes, in it until the end until he returned to the Mets. Although Cespedes is right-handed, the power numbers were too attractive to pass up.

Next came a two-year offer to Dexter Fowler, who turned down three years from the Orioles at about $35 million as well as the White Sox’s proposal and returned to the Cubs for one season. Andre Eithier of the Dodgers was also pursued before spring training began.

The White Sox may now be focused on Rockies outfielder Carlos Gonzalez and Reds outfielder Jay Bruce.

Gonzalez hit 40 homers and had 97 RBIs last season. The money for Gonzalez and what the Rockies front office is asking for in return appear to be the holdup right now.

“They asked for my top two minor league players, a major league player and to absorb most of the contract,” said a general manager who was interested in Gonzalez. “We will wait for something more realistic ”

Gonzalez has $37 million left on his deal over two years.

What might be more practical for the White Sox? A deal for Bruce, who’s 28 and making $12.5 million in the last year of his deal before becoming a free agent. Considered a solid all-around player and clubhouse influence, Bruce has averaged 26 home runs and nearly 80 during his past eight years in Cincinnati. The troubling numbers include a huge slide down to a .217 batting average in 2014 and a .226 mark in 2015 while averaging nearly 160 strikeouts over the past three years.

Former teammates Navarro and Frazier like what Bruce brings to the field.

“Great teammate and great guy,” Navarro said of Bruce. “He is one of those guys that is going to go out there everyday and give you his best.If we get him he would be a huge plus for us.”

Added Frazier, a teammate of four years: “He was a great teammate for me. He got into the major leagues as a young kid. He developed into a good ballplayer. He goes about his business the right way. He works his tale off.”

With an aggressive owner who wants to win every season, White Sox general manager Rick Hahn and executive vice president of baseball operations Kenny Williams can continue their pursuit of a big run-producing bat from the left side on a daily basis.

Bruce Levine covers the Cubs and White Sox for 670 The Score and CBSChicago.com. Follow him on Twitter @MLBBruceLevine.