FOXSports.com first reported that the White Sox wanted Ramirez -- currently on the disabled list for the third time this season and not expected back for another week or more -- but were willing to pay only $1 million of the more than $7 million remaining on his contract and would not give up any players. The Dodgers reportedly asked for Dayan Viciedo.
The Rays and Angels also reportedly have contacted the Dodgers about Ramirez.
Manager Joe Torre has insisted that Ramirez remains a productive presence in Los Angeles' lineup when he's healthy, even though he no longer possesses the power that made him one of the most feared hitters in the game. He has eight home runs in 61 games and a .317 average.
General manager Ned Colletti said he received several calls on Ramirez on Saturday, but "it was too cumbersome to get that accomplished" with Saturday's 1 p.m. PT Trade Deadline bearing down. The White Sox apparently contacted the Dodgers on Friday.
"We got a call from one club, and they offered us a very low dollar figure with no player attached to it," said Colletti. "We didn't start it and we didn't float his name. One team came forward, and I'm not sure what they were trying to accomplish."
The GM said he expects Ramirez to remain with the Dodgers.
"At the end of the day, we still need Manny, we need him in the middle of the order and we need him to get back sooner or later," said Colletti.
"The club was built with him and is best with him as the left fielder. Because we haven't had him for much of the year doesn't mean he can't provide us some help the rest of the season. If another club had brought us a player back that helped us win, we'd probably consider that, if it worked for Manny, as well, with the no-trade clause."
White Sox general manager Ken Williams has always been known to think big where improving his team is concerned. He's talked in the past about calling the Twins concerning Johan Santana before the left-hander was traded to the Mets and hinted about offseason pursuit of Roy Halladay.
"That's an impact type guy, so you can reasonably assume we kicked the tires on it but obviously we didn't kick them hard enough," said Williams.
"Again, that was something, since enough of it is out there and they've confirmed and spoke on it, I'll just say we didn't get any sort of dialogue back and forth because they were involved in so many other things that it wasn't a high priority."
Although the White Sox were thought to be looking for a left-handed bat to balance a right-handed-heavy lineup, manager Ozzie Guillen has talked about just wanting a productive hitter regardless of what side of the plate he hits from.
The 38-year-old Ramirez has a no-trade clause, but he has told teammates he'd prefer to be a designated hitter to preserve his legs. Ramirez has missed 40 percent of the season with a pair of calf muscle strains and a hamstring strain, all in his right leg.
Concern over Ramirez's health, and that of sidelined outfielder Reed Johnson, led the Dodgers to acquire Scott Podsednik this week from the Royals.