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05/30/10 2:55 PM ET

Could Konerko fit in Anaheim?

ST. PETERSBURG -- A freak injury suffered by Angels first baseman Kendry Morales during Saturday's celebration following his walk-off grand slam already has pundits guessing as to whether White Sox first baseman Paul Konerko could be a replacement target for Los Angeles. Konerko has full no-trade veto power, by virtue of at least 10 years in the Majors and at least five straight with the White Sox, but if the White Sox are unable to fight their way back into the American League Central race, he could choose to go to a contender if the option is presented.

Konerko's Arizona home is not far from the Angels' regular-season home or where they train during the spring in Tempe. The White Sox team captain also entertained a serious offer from the Angels when he was a free agent after the 2005 championship season and has a connection to Anaheim manager Mike Scioscia from their days with the Dodgers organization.

"The Angels have been looking for Konerko for a long time," said White Sox manager Ozzie Guillen. "That's not my call. Hopefully Konerko is playing good and then like I say, it makes [the front office] make a decision."

Guillen mentioned how he has spoken with bench coach Joey Cora in the past concerning players possibly getting injured during walk-off celebrations. Morales suffered a broken left leg as he jumped toward home plate following his game-winning blast.

"I'm not against people celebrating," Guillen said. "Just be careful how you celebrate. Hopefully, players will learn from this example about how you are going to celebrate."

June to determine fate of White Sox

ST. PETERSBURG -- The calendar says June begins on Tuesday, when the White Sox play host to the Rangers at U.S. Cellular Field. But in White Sox manager Ozzie Guillen's mind, the next few weeks mean far more than a change to warmer weather for the struggling White Sox.

With the non-waiver Trade Deadline coming up on July 31, June could be the time for the White Sox to make a truly positive statement or continued subpar statement where the 2010 season is concerned.

"It's going to be a month that is going to show [general manager] Kenny [Williams], 'OK, we add or we subtract,'" Guillen said. "What way are we going to go? Or we going to move people or add people to help?

"That's what I call this month. I want to keep those guys, but in the meanwhile, if we keep playing like that, Kenny has a job to do and [White Sox assistant general manager] Rick [Hahn] have a job to do. They have to do what they have to do.

"We have to play good for them and make it tough for them to make decisions," Guillen said. "Right now, the way we are playing, it's easy to make decisions. We have not performed the way we thought. When you play good, make them think about it and say, 'Wow. What are we going to do?' That's what I expect the players to do to make this month very good and hopefully they can stay good."

In 2007, Guillen's worst year of his seven running the White Sox, it was evident by late May that the team didn't have much of a chance to contend. In '08, Guillen understood that the team would have to fight all the way to the end to reach the playoffs, and actually fought through an extra American League Central tiebreaker as Game No. 163.

This year's group, though, has been the most disappointing for Guillen, in that it goes against the success he expected when the team broke camp from Arizona.

"Yes, with no doubt," Guillen said. "I couldn't care less what people think and say and what the expectations are. I know as soon as we got this ballclub. I remember telling Kenny 20 times, 'I like what we have. I think we are OK. Don't add any more players.'

"He kept calling me every week. Well, I guess Kenny knew more than I do because he said he wanted to add someone else and I was,'No, no, no.' Because looking at this lineup and this team, this team is not a .500 team or even below .500 team. I don't think so. I don't want to say it's a shame because things happen and we still continue to play, but it surprises me the way we are playing."

White Sox in search of the long ball

ST. PETERSBURG -- Andruw Jones entered Sunday's series finale at the Tropicana Dome sitting at 397 career home runs. He has been in that same holding pattern over his past 52 homer-less at-bats.

Paul Konerko has three long balls in May, after blasting out 11 in April, while Carlos Quentin went deep for the first time in May on Friday night off Tampa Bay's Wade Davis.

This temporary lack of power has surprised White Sox manager Ozzie Guillen.

"I'm surprised because the weather is better," Guillen said. "Right now, Andruw is trying to pull everything, he's not staying in the middle of the field like he was. I think he's maybe trying to hit home runs; PK the same way.

"Yeah, it's been a surprise. We've played a couple games at home where we thought there were going to see some home runs. But like I say, we're not a home run hitting team. If we try to hit the home run, then we're done. If we try to hit the ball pretty good, we have enough guys with power that can hit the ball out of the ballpark."

Guillen doesn't want his team to get home run happy as summer approaches in Chicago and with the White Sox playing all but six games in Chicago until June 28.

"Everybody has to take the approach that not just because we're in Chicago we're going to turn everything around and be like a powerhouse," Guillen said. "We have to play good at home. There's nothing worse than when your own people are booing you and they discredit you and they criticize you."

Thornton likes pitching in dome

ST. PETERSBURG -- The White Sox finished their four-game 2010 ledger at Tropicana Field on Sunday, and count reliever Matt Thornton as one visiting pitcher who doesn't mind pitching in the dome.

"I've always liked when I'm actually in a game in a dome," Thornton said. "I like the atmosphere. No rain, no wind, no hot or no cold. It's perfect to pitch in.

"Sitting here beforehand in the daily routine, that I hate. I hated it in the Metrodome, the Skydome and here. But pitching is not bad at all."

Scott Merkin is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.