Rumor roundup: Brandon League, Jonathan Broxton, James Shields

Here's the latest on Brandon League, Jonathan Broxton, and James Shields:

• League is close to being traded, tweeted ESPN's Jerry Crasnick, but not to the Rangers or Braves, according to sources, he added. 
• League, 29, has been great in July: he's allowed two earned runs and seven hits in 9.2 innings, good for a 1.86 ERA and 1.03 WHIP. He's struck out five and walked three over that span.
• The White Sox have been having discussions with the Mariners about League, tweeted CBSSports.com's Danny Knobler.
• The White Sox have been one of the more active teams at the deadline, adding Francisco Liriano in their most recent move on Saturday night. When general manager Kenny Williams was asked if more trades could be made, he answered with a definitive, "Yes," wrote the Chicago Tribune's Mark Gonzales. 
• The Giants aren't about to land League, tweeted the San Francisco Chronicle's Henry Schulman. He noted that the Dodgers are in the mix.
• The Giants are instead focusing on Royals reliever Jonathan Broxton, tweeted ESPN's Jim Bowden. 
• There's a 60-to-70 percent chance Broxton is dealt by Tuesday, tweeted Crasnick. 
• Earlier in the week, Knobler tweeted that the Giants didn't view Broxton as the answer in seeking bullpen help, but the relief market is awfully thin, and only getting thinner. 
• Rays manager Joe Maddon reiterated on Sunday that he doesn't anticipate James Shields being moved before Tuesday's deadline, tweeted CBSSports.com's Scott Miller.
• The Rangers are believed to be far more interested in Shields than Josh Johnson, Miller added, also on Twitter.
• Rival executives maintain, however, that they believe Shields will be dealt before Tuesday's deadline, wrote ESPN's Buster Olney. 
• Here's what Rays executive VP Andrew Friedman had to say: 

  "We've been having a lot of different conversations with a lot of different teams, which is very typical for this time of year. Our policy of not commenting publicly provides endless humor-filled fodder. A lot is made of things that are just not true. As always, we are looking for ways to positively impact our organization and make it better in the short term as well as the long term. If we're able to line up on a deal that does that, we're going to be very aggressive. As is customary this time of year, a lot of different things are being discussed, so it is difficult to accurately handicap the chances of a deal."