FoxSports.com first reported the story.
The source said Ramirez would become the White Sox's designated hitter against all pitching, and that going back to the American League, where he wouldn't have to play the outfield, appeals to Ramirez.
Ramirez said he's heard nothing amid the speculation.
"No, man, nothing," Ramirez said while snacking on a banana before returning to a conversation in Spanish with teammates Vicente Padilla and Ronnie Belliard. The Dodgers clubhouse closed moments later for a standard team meeting before Los Angeles began a three-game series with Milwaukee on Tuesday night.
If the Dodgers delay putting him on waivers, it could impact the Sox's decision due to their position in the standings.
The Dodgers most likely will put Ramirez on waivers Tuesday or Wednesday in hopes of ridding themselves of an unproductive player who no longer can play the outfield sufficiently in the National League. Ramirez, 38, has been on the disabled list three times in 2010.
"I don't think that it's anything different than what goes on in July. All the rumors that fly around in July about trade possibilities," Dodgers manager Joe Torre said. "You know I can't talk about what's going on, but just the attitude, I don't see any concern, first off in his game. ... You really are bred to deal with distractions. It's our job, and so, I don't see it affecting him at this point."
White Sox manager Ozzie Guillen said if he was asked by general manager Ken Williams if he wanted Ramirez, he would say yes, even though Ramirez's value is a question.
"Manny is a tremendous hitter," Guillen said. "I don't think anybody doubts that. Can he help us? I don't know because he hasn't played in the big leagues in a while.
"You talk to me about Manny and I didn't even see Manny in spring training and we played against him every day. I don't know. If you give me Manny when he was with Boston, I take a limo to pick him up. But I haven't seen Manny in a long time."
Ramirez came into Tuesday night's game against Milwaukee hitting .312 with eight homers and 39 RBIs in 62 games.
Williams said it's a violation of tampering rules if he spoke specifically about Ramirez, but said that his club can never have too much offense.
"We've got to have the conversation and see what the numbers are on the salary, how we're doing in the standings and the gate. The last 30-some odd days it changes seemingly on a day-to-day basis," he said. "You have to think really long and hard about where you are at a given time and what you're going to do."
The waiver procedure goes in reverse order of standings, starting with the player's current league. So in the case of Ramirez, the team with the worst record in the National League has the first opportunity to put in a claim. It's perceived that no team in the NL envisions Ramirez as an every-day position player.
As far as AL contenders go, the White Sox have the worst record. Therefore, it's perceived they will have the first shot at Ramirez before teams like Texas and Tampa Bay, who may have their own interest in Ramirez.
If and when the Dodgers place Ramirez on waivers, they a have a chance to either receive a player or players back from the team that claims him, eat a portion of Ramirez's contract, or let the claiming team take Ramirez with no compensation. That team would be obligated to pay Ramirez the $4.25 million remaining on his 2010 contract.
The Dodgers also can pull Ramirez back, meaning he won't be able to be on waivers for 30 days, rendering him ineligible for postseason play. Players must be on the roster by Aug. 31 to be available for the postseason.
Ramirez has a no-trade clause in his contract, meaning he must approve the deal before any transaction is finalized.
Dodgers owner Frank McCourt was at Miller Park for the unveiling of a statue honoring baseball commissioner Bud Selig, as was Reinsdorf. McCourt said the day was about Selig and he wasn't discussing any business.
A Dodgers spokesman said GM Ned Colletti wasn't on the team's charter to Milwaukee, but is expected to join the team at some point during this series.
The White Sox believe Guillen and his coaches can easily communicate with a player like Ramirez, and find it appealing that Ramirez has hit .325 vs. right-handed pitching and .257 vs. left-handers.
According to the source, the Sox are prepared to assume the $4.25 million owed Ramirez off his original $20 million salary for 2010. Ramirez is a free agent after 2010. Any claiming team would not have any obligation to him after this season.
With 16 home games remaining, the White Sox could easily recoup a large portion of Ramirez's contract via new ticket sales and marketing. Owner Jerry Reinsdorf has shown in the past he's not afraid to spend money and make a big splash if the decision makes baseball and economic sense.
The Sox most likely won't have to give up a player in return, unless the Dodgers want to pay some of Ramirez's salary, which isn't likely.
Several White Sox players were intrigued by bringing in Ramirez to help bolster their offense.
"It's just a rumor at this point. If he shows up tomorrow in his uniform, we'll all welcome him," Paul Konerko said. "Hopefully it'll be good. But until then, it's tough to comment on. We only see as players what you guys see, what's on the TV, if at all."
Information from ESPNChicago.com's Jon Greenberg and The Associated Press was used in this report.