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7/22/2014 9:10 P.M. ET

Danks subject of trade rumors near Deadline

White Sox lefty would need to approve deals with Blue Jays, A's, Nats, O's

CHICAGO -- It was a little less than three weeks ago when White Sox starter John Danks stated that trade rumors involving his name before the July 31 non-waiver Deadline weren't going to occupy any part of his daily routine.

That philosophy hasn't changed, even with Jon Heyman of CBSSports.com reporting on Tuesday that the Yankees have checked in on the veteran southpaw.

"Yeah, nothing changes. It is what it is," Danks told MLB.com before Tuesday night's game with the Royals. "It's part of it. Something happens, we'll deal with it then."

Danks had a full no-trade clause in the first year of his five-year, $65-million extension, but he had a limited no-trade clause for six teams where his approval is needed over those remaining four years of which two years at $28,500,000 remain after the '14 season. A Major League source confirmed that four of those teams are current contenders: the Blue Jays, A's, Nationals and Orioles.

While Danks wouldn't confirm the no-trade list or really had any desire to talk about trade rumors in general, he did say that teams being on that list doesn't mean he would automatically deny a deal if that particular team showed interest.

"It certainly would be something I would sit down with [his wife] Ashley and kind of weigh the pros and cons," said Danks, who heard about this particular rumor via a text from his friend. "But I hadn't really given much thought to it."

Through 20 starts this season, Danks is 8-6 with a 4.35 ERA. He had a rough performance on Sunday against the Astros, but he has recorded eight quality starts over his last 11 outings. His next start is scheduled for Friday against the Twins at Target Field.

Eaton receives Sox's Heart and Hustle award

CHICAGO -- Adam Eaton did not want to talk on Tuesday about a Chicago Sun-Times report that the White Sox leadoff man was playing with a broken middle finger on his non-throwing hand. He was more than happy to discuss his honor as the club's 2014 Heart and Hustle Award winner as announced on Tuesday by the Major League Baseball Players Alumni Association.

One player from each Major League club was selected for his passion for the game of baseball and who best embodies the values, spirit and tradition of the game. This award is voted on by former players, with the overall winner to be announced on Nov. 18.

"I'm pumped. My wife and I talked about it today, and it's my first big league trophy, which is kind of cool," Eaton said. "I'm honored. I remember Willie Bloomquist getting it last year when we were in Arizona, and for me it's something you should get every year.

"If you're playing the right way and healthy, and doing what you should be doing, you should definitely compete for that trophy every year. I'm very pleased, and I feel like I'm in with some good company. You've got to play hard every day, and I'm blessed to have the opportunity, and I'm excited."

Eaton entered Tuesday's contest against the Royals with an eight-game hitting streak, and was 7-for-10 with three walks over his previous three games. So whatever the issue is with his finger it doesn't seem to be a problem.

"I'm fine. It's not broke. It's just hurt," Eaton said. "I'm fine. I don't even want to talk about it. Let's not talk about it. We're good."

Abreu held out of lineup with back stiffness

CHICAGO -- White Sox manager Robin Ventura held Jose Abreu out of Tuesday's lineup due to back stiffness that he noticed while Abreu was playing on Monday against Kansas City.

"Last night, I was watching him run around. Going through the All-Star stuff and everything else not having a break, you just see little things, back stuff and him moving around," Ventura said. "I talked to him last night after the game and just being honest with him, and him being honest with me. So today was just a good day to do it."

Abreu has hit in 12 straight games and 30 of his last 31, while sitting on pace for 47 homers and 120 RBIs. The rookie first baseman missed 14 games earlier this year due to a left ankle issue, but he is expected back in the lineup for Wednesday afternoon's finale.

"His ankle has been fine. This is just more of you can see him stretching his back and doing some moves that you know it doesn't quite feel right," said Ventura, who mentioned Abreu was available to pinch-hit on Tuesday. "It's just more to give him a day.

"We talked about this a few days ago that every once in a while when things come up, just knowing that he's starting to get to the most games he's ever played in a year portion. When things like this happen, you just give him a day."

Reinsdorf praises La Russa's Hall induction

CHICAGO -- When chairman Jerry Reinsdorf bought the White Sox in 1981, one of the first things he thought that he was going to have do was fire then manager Tony La Russa because the television broadcasters, Harry Caray and Jimmy Piersall, kept talking about how bad La Russa was at running the ballclub.

"Then, I met him and realized how wrong they were," Reinsdorf said on Tuesday. "Obviously having two people connected with the White Sox going in [the Hall of Fame] at one time is special, but it's really special for me to see Tony go in knowing how he suffered early in his career and the abuse he took, and to see that he proved all the critics were wrong.

"I just wish Harry Caray were alive," Reinsdorf added.

The Hall of Fame induction ceremony will air Sunday at 1:30 p.m. ET live on MLB Network, simulcast on MLB.com and the At Bat app.

Reinsdorf described his relationship with La Russa as "becoming like brothers" over the years, adding that as great of a manager as La Russa is, "he's a better human being. Just a great person."

One of Reinsdorf's greatest regrets as an owner was eventually letting go of La Russa, after current TV broadcaster Ken 'Hawk' Harrelson took over as the team's general manager in 1985. La Russa and Harrelson had a difference of philosophy during the '86 season, per Harrelson's comments on Tuesday, and Reinsdorf believed that someone running a department should have his own people in place.

That in-season dismissal didn't happen, though, before Reinsdorf called Roy Eisenhardt, the president of the A's, and set up La Russa for his next managerial gig. Jackie Moore was fired as Oakland's manager six days later, replaced in the interim by Jeff Newman. La Russa came in three weeks after the Moore firing.

"It was the biggest regret [letting him go]," Reinsdorf said. "Well, it was the combination, naming a general manager that shouldn't have been a general manager and then letting him fire Tony."

"He managed what 33, 34 years in the big leagues, and it's funny he only got fired one time, and you're talking to the [person] that fired him," Harrelson said, smiling. "He might go down as certainly one of, but maybe the best manager we've ever seen."

Third to first

Tyler Flowers just missed a three-run homer during Monday's 3-1 win over the Royals, but he made it out of the ballpark in time to be with his wife Nancy shortly after the contest for the birth of their son, Brantley Cole. The newest Flowers checked in at 6 pounds, 11 ounces. Flowers did not start on Tuesday, but he was available off the bench.

Matt Lindstrom remains on track to throw live batting practice to Avisail Garcia this weekend in Minneapolis. Lindstrom, who is recovering from an ankle injury, could depart for a Minor League rehab assignment by Aug. 3, according to Ventura.

• Advanced Rookie affiliate Great Falls won, 12-8, at Idaho Falls on Monday, clinching the Pioneer League North first-half title. The playoff appearance will be the eighth straight for the Voyagers.

• There still is no timetable for the return of pitching coach Don Cooper, who has been sidelined by vertigo.

Scott Merkin is a reporter for MLB.com. Read his blog, Merk's Works, and follow him on Twitter @scottmerkin. This story was not subject to the approval of Major League Baseball or its clubs.