The Chicago White Sox and Los Angeles Dodgers were popular preseason picks to contend for the World Series.

As the teams began a three-game series Tuesday at Guaranteed Rate Field, the Dodgers have indeed started strong and lead the National League West in early June.

The Sox, however, began the day in third place in the American League Central and two games under .500.

General manager Rick Hahn said the first couple of months have tested his patience.

“But I think that makes me no different from any White Sox fan or ardent follower of this club,” Hahn said before Tuesday’s game. “We’ve all been tested over the last few months here. I get the benefit of being inside those meetings and those conversations, being able to see the progress on some of the injured players, being able to see some of the things we’re working on to get some of the guys who have struggled out of their ruts.

“So I’m able to have not only the ability to address some of these things personally, but also feel like the reasons for optimism are legitimate and make me feel better and make me give it a little longer, perhaps, before my patience runs out.”

Hahn added: “All of us — whether it’s (Chairman) Jerry (Reinsdorf), (executive vice president) Kenny (Williams), myself, the coaches, any White Sox fan — we’ve all had our patience tested. But the fundamentals of who this team is remain. We’re fortunate that baseball is a long season, and over the course of a long season, things tend to play out the way the talent permits. And we feel good about this talent.”

The offense has been inconsistent. And for the second straight season, injuries have hit the Sox hard.

Left fielder Eloy Jiménez (torn right hamstring tendon) and starting pitcher Lance Lynn (right knee surgery) are rehabbing at Triple-A Charlotte.

“(Jiménez is) progressing well physically,” Hahn said. “It’s really, frankly, a matter of getting his legs completely under him and getting his timing at the plate and rhythm back. I don’t have an exact time to project his return, but it’s getting closer, knock on wood.”

Lynn is scheduled to make his third rehab start Wednesday. The plan is for five innings and between 70 and 80 pitches.

“If that goes smoothly, we’ll then have a conversation about next steps for him, which could well include being activated in Chicago,” Hahn said.

Shortstop Tim Anderson has been doing drills with the Sox to help with his recovery from a right groin strain.

“Tim had a good weekend in Tampa in terms of his healing and progression toward rehabilitation,” Hahn said. “I don’t have a timeline just yet on a potential rehab assignment for Tim. If things continue on this path, hopefully sometime when we’re on the next road trip we’re able to get Tim on a rehabilitation assignment and then go from there for his return.”

Anderson went on the injured list retroactive to May 30.

“I believe the original projection on him was three weeks,” Hahn said, “and we remain on that path.”

Hahn said pitchers Joe Kelly (left hamstring strain) and Vince Velasquez (left groin strain) might be in position to throw off a mound during the homestand with the possibility of both being activated by the end of the month.

“I feel like we’ve endured what will hopefully be our toughest part of the schedule, certainly from an opponent standpoint,” Hahn said, “but more importantly from a manpower standpoint where we have been from, ‘Who’s been available?’ We expect that to improve here over the coming weeks.

“Certainly there are segments of the past two months that we can be proud of. Going 11-8 against the (New York) Yankees, (Boston) Red Sox and Tampa (Bay Rays) is pretty good and something that certainly answers some of the concerns about our ability to match up with good teams.

“Obviously there have been some stretches that have been pretty disappointing, ruts that — on the positive side we’ve been able to pull ourselves out of — but on the negative you don’t ever want to endure. I think right now in that clubhouse and that coaches room, there’s a lot of optimism and excitement to get some of these guys back and build on some of the momentum from this past weekend.”

The Sox won two of three against the Rays at Tropicana Field. The Dodgers provide another test. Sox manager Tony La Russa said “yes and no” when asked whether he looks at the series as a measuring stick.

“(The) yes is (the Dodgers are) really good,” La Russa said. “When we’re at our best offensively, we do what they do. Very aggressive, go a lot at the strike zone, just put the ball in play, don’t strike out much. And that’s when we’re at our best, so hopefully that’s what we show.

“The no is whatever we do to get ready, we better do the same against Texas (in the next series). Because you’ve got to get wins in this league. Whoever you play, you’ve got to get wins. We want to have a real consistent will for it every day. And they’re capable of doing that.”

()