By Bruce Levine–

NASHVILLE, Tenn. (CBS) — The White Sox aren’t done adding offensive power to the infield mix.

After trading two young pitchers for Athletics infielder Brett Lawrie on Wednesday night, the White Sox have more moves up their collective sleeves.

The team has added Lawrie, who can play second or third base. With 16 home runs and 60 RBIs in 2015, the 27-year-old Canadian can be projected in a hitter-friendly park like U.S. Cellular Field for 20 or more long balls next season.

“We talked about the heightened expectation to get something done here at these meetings,” general manager Rick Hahn said. “There is a fever pitch when you are at the meetings. You want to address needs. We talked about this before. There was no urgency to do this one here. We were able to close on it. Now it’s on to the next one.”

That next one Hahn is talking about might play the hot corner for the Cincinnati Reds. Third baseman Todd Frazier is available in the right deal for some top prospects.

The Reds are in a salary-dump mode, and Frazier — who hit 34 home runs last season — will earn a modest $7.5 million in the last year of a two-year, $12-million deal. Frazier will be offered salary arbitration in 2017 and then be eligible to become a free agent in 2018.

The Reds want either right Fulmer, the White Sox’s first-round pick in 2015, or top minor league shortstop Tim Anderson for Frazier. The teams have had at least one conversation with these names being bandied about. That price appears too steep for Chicago right now.

With the Lawrie addition, the White Sox could go the international route for an alternative third base candidate.

The White Sox have interest in 32-year-old Japanese third baseman Nobuhiro Matsuda, who would like to play in the U.S. on a short-term deal, according to an international source. After Matsuda hit 35 home runs in Japan last season, the short porches at U.S. Cellular Field might be a perfect landing spot.

The contract for Matsuda would be reasonable by American standards, because there are no posting fees for Japanese players who have 10 years or more service time in Japan.

Chicago still has a sufficient amount of payroll dollars to add a free agent or another veteran through trade. With Lawrie making around $4 million next season, the White Sox are now at around $105 million in payroll, projecting arbitration players as well. The front office will likely be able to go to around $120 million, similar to the payroll figure from 2015.

“We want to keep adding and improving the offense,” Hahn said after trading for Lawrie. “We are pleased with the improvements we made thus far. In our mind, we hope to continue. We don’t think we are done.”

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