CHICAGO -- White Sox general manager Rick Hahn was not about to talk about any specific player potentially holding interest for his team as the Aug. 2 Trade Deadline approaches.

That edict included players on other teams, of course, or even a player on the open market such as outfielder Michael Conforto. So, when Hahn met with the media Friday afternoon and was asked about a player of Juan Soto’s excellence and expected high salary point to extend holding interest, he took a roundabout way of still making his point.

“How should I put this?” Hahn said. “I will say historically, as long I've been here working for [White Sox chairman] Jerry [Reinsdorf] and [executive vice president] Kenny [Williams], there hasn't been a premium player that changed teams that we haven't at least had a conversation about.

“You were probably, and this is my fault, we may have been a little too forthright or upfront in very recent premium free agent pursuits that led to fans being frustrated when we didn't wind up converting on them. So I think at this point, I'm going to err on the side of that White Sox stealth mode … and if we make the deal, I'll happily explain how it all came together.”

That premium free agent list would include third baseman Manny Machado, right-handed starter Zack Wheeler and outfielder Bryce Harper, players the White Sox pursued at some level over the past few years but eventually went elsewhere. Any team acquiring Soto, if the Nationals decide to move him, would have a little more than two years of control before he entered free agency.

All of this Soto talk is primarily hypothetical at this point. But Hahn did speak Friday about potential changes for a team entering the start of the second half trailing the Twins by three games in the American League Central. They were centered on areas of no surprise.

“Every team in baseball, whether they're in it or not, never feels like they have enough pitching and we're certainly not immune to that, primarily in the bullpen,” Hahn said. “Obviously, Aaron Bummer has been absent for a while. He's starting to make some progress and we do project a return at some point, but that's not guaranteed.

“So I would say bullpen's probably the most obvious need. I think if you look at our offensive and defensive performance over the course of the year, the performances at second base and right field haven't quite been what we would hope. That said, if you look at what Josh Harrison has done since June 1, and you look at what Gavin Sheets has done since he returned from the Minors, those both potentially could be solutions there. We'll have to wait and see.”

The White Sox are a stronger offense against left-handed pitching, so a left-handed bat would help. Hahn wouldn’t even rule out a starting pitching addition, although the White Sox are pretty well set with Lucas Giolito, Dylan Cease, Johnny Cueto, Lance Lynn and Michael Kopech, even with Kopech’s workload being watched as the season progresses, and Hahn doesn’t see a great deal of volume in that market.

Hahn also won’t wait to make a move, gauging if a team currently in contention falls out in the next few weeks, if the deal makes sense and makes a difference.

“If we find the right deal for a player we think is impactful, we are going to want to pull the trigger on it when it’s available to us and not wait around,” Hahn said. “Take that bird in the hand vs. the two in the bush.

“With that said, it strikes me that more teams being in it, the added playoff spots, that a lot of these decisions might come down to the Deadline so teams know exactly where they sit. ... The chatter has ramped up since Monday. I can’t tell you it’s because the Draft is over and we can only focus on one thing at a time as much as we’re within a couple weeks of the Deadline, so it’s only natural teams ramp up.”