If the White Sox are competing against the Cubs, Yankees, Dodgers, Red Sox, Giants, Astros, they are not likely to land that player unless they go well beyond the expected offers and just make an offer that the player would not say no too. I don't think it's likely a player will take a similar offer with the White Sox to grow with the young core if other premier franchises are in the running.
I could see them looking at trade options just so they can land a key player who can help next season.
I don't know if they will land the top of the free agent class, but I do expect that they will be in the market for a RF/DH (maybe both, but doubtful). They will be looking at the bullpen, but I don't understand the logic that had them giving Herrera the deal (2 years, 18 M) he got when his numbers were declining.. Still, I expect to see another veteran added there.
I do think they need a #1 or # 2 type starter, but will this come from free agency or trade? Not sure if it's free agency if the teams mentioned are in on Cole.