CHICAGO -- Jesse Crain played catch Tuesday for the first time since being put on the disabled list, retroactive to April 21, with a strained left oblique. The next step for the right-handed reliever is to throw a side session Thursday.

"He still feels some stuff," said White Sox pitching coach Don Cooper. "He feels it a little bit, but it was nothing. I don't think we've lost any time."

Crain is eligible to come off the disabled list Sunday in Detroit. But there could be an injury rehab appearance before Crain's return.

"We haven't discussed it much further than that as to do we do it that day, do we send him out?" Cooper said. "Is he going to be OK after these two sidelines to go back into a Major League game or does [White Sox general manager] Kenny [Williams], [assistant general manager] Rick [Hahn] combined with [head athletic trainer] Herm [Schneider] think it's better off for him to have an outing somewhere else? We haven't gotten there yet."

Ventura remains patient with Beckham, Morel

CHICAGO -- Support for struggling White Sox infielders Gordon Beckham and Brent Morel continued to come in Tuesday from manager Robin Ventura.

But as the calendar turns from April to May and with Morel (.182) and Beckham (.190) sitting a few games of consistent hitting below the Mendoza line, even following Beckham's three-hit effort in Tuesday's 7-2 victory over the Indians, that unconditional support from Ventura soon might be slightly altered.

"Eventually you probably have to do something," said Ventura before the series opener with the Indians. "Right now I'm believing what I see and talking to them.

"I want to know what they're thinking, too. I'm still confident they're going to perform. Eventually it gets to the point where you would have to make a move if it would stay like that, but right now I'm not going to do that."

A few factors work in favor of the Beckham/Morel tandem staying put, starting with their stellar defense. There also doesn't appear to be a clear-cut favorite for everyday replacements, although Ventura believes options exist.

"Yeah. I think so," Ventura said. "But I haven't looked that far down the road."

Brent Lillibridge and Eduardo Escobar have the ability to play both second and third, but they make the team stronger in a utility role. Ventura requested an extended Spring Training look at Tyler Saladino, who was then promoted to big league camp and is now hitting .253 as the shortstop for Double-A Birmingham. Drew Garcia, the club's 21st-round selection in the 2008 First-Year Player Draft, is hitting .316 as a middle infielder for the Barons, while Tyler Kuhn could be a candidate at Triple-A Charlotte.

All of these names are pure speculation, with a corresponding roster move needed to add any of them. Ultimately, the best team option is for Beckham and Morel to stop pressing and start hitting, as they showed signs of on Tuesday.

"They do a lot of good things defensively, and for me, that's a plus," said Ventura, who had a much-publicized 0-for-41 stretch in his rookie campaign. "It's not easy to hit, and it's especially not easy to hit when every little thing is scrutinized. They have to get to that point that they don't care about that and it'll turn around for them."

"I want them to do well," said Lillibridge, who simply tries to be ready to play and not analyze where opportunities might occur. "Their confidence has to be there, and hopefully this series, a new month, however you want to break it down, gets them right and they start swinging the bats. We need them at the bottom to get on base for the guys swinging the bats well and carrying this team."

Reed approaching club record for rookie hurler

CHICAGO -- Addison Reed's 10 straight scoreless appearances, covering 8 1/3 innings, ranks as the third-longest run by a White Sox rookie reliever at a campaign's outset since 1921, according to STATS LLC. Sergio Santos holds the mark at 12 in 2010, followed by Matt Karchner with 11 in 1995.

That record hasn't crossed Reed's mind -- literally, not for a second.

"The only time I've ever thought of it is right now," said a smiling Reed.

When asked if he even knew about the record, Reed quickly admitted that he had no idea.

"It's not going to change anything. Every time I go out, I try to not give up any runs," said Reed, who has fanned 11 and walked three. "If it happens to be a record, then it happens to be a record.

"I'm just going out there trying to throw up zeroes and help the team win. So, I'm not thinking anything of it."

Reed has proven to be a late-inning force, while rookie Nate Jones has earned a late-inning look with Jesse Crain on the disabled list because of a strained left oblique. Closer Hector Santiago and middle reliever Zach Stewart complete this young nucleus of White Sox relievers learning on the fly with a 2.93 bullpen ERA.

"We are staying hard on [Jones], Hector, Reed and Stewart," White Sox pitching coach Don Cooper said. "We are making sure that everything they are doing is trying to build a solid foundation we are going to need to help us now as well as in the future."

Lillibridge in elite but odd company

CHICAGO -- Entering Tuesday's series opener with the Indians, Brent Lillibridge was one of two players in Major League Baseball to have more steals than hits. Lillibridge has six stolen bases and four hits, while Baltimore's Nick Johnson was hitless with one stolen base.

The White Sox utility player, who had just 21 at-bats over 17 games played, doesn't believe his team-best 6-of-6 on stolen bases will continue without drawing attention from opposing teams.

"They haven't pitched out at all on me," Lillibridge said. "At one point, it's going to start coming so I just have to be smarter. For the most part, I'm getting good chances to run knowing it's going to be a certain pitch. I just have to keep working on it, help whenever I can and wait for an opportunity to secure a role."

Lillibridge had a breakout year in 2011 as the top White Sox defensive outfielder and with 13 homers and 10 stolen bases. He admitted to feeling a little uncomfortable at the plate over a couple of games but has figured out a little something with his swing to give him a stronger feel.

"I want to be here for a long time," Lillibridge said. "I know I'm worth something, and I feel like I have a great opportunity to help this team one way or another."

Third to first

• Robin Ventura reiterated Tuesday that Hector Santiago remains his closer, despite Matt Thornton closing out Sunday's victory over the Red Sox. Santiago pitched a scoreless ninth in Tuesday's 7-2 victory over the Indians, marking his first appearance since last Wednesday.

• The White Sox 11-11 April finish raised their opening-month record to 41-53 since 2009, still the third-worst winning percentage in the Majors behind Baltimore and Houston.

• Tuesday's win gave the White Sox a 5-1 record vs. the AL Central.