In fact, Williams indicated Thursday evening how there might be a candidate other than the obvious or most talked about currently being pursued by his team.
"It's possible," Williams said with a big smile. "It's possible. It's always possible.
"Sometimes the most obvious things are the most unrealistic things to accomplish, so you always have to be looking for what you actually can accomplish. And sometimes small marginal growth is just as effective as the more obvious bigger thing that you're not going to get."
Why wouldn't the White Sox get Ramirez? For starters, a National League team or an American League team with a lesser record than the South Siders also could put in a waiver claim, and the priority would go to that team.
Another possibility falls upon the Dodgers deciding to keep the talented right-handed hitter, beset by injuries during the course of the 2010 season. Ramirez has just two hits and an RBI since July 17, but he has only played in three games during that same time frame.
Los Angeles has crept back into the NL Wild Card race, entering Thursday night's slate of action trailing San Francisco by a mere five games. The Dodgers must jump the Rockies, Cardinals, Phillies and Giants, but the latter three have not played well of late.
Ultimately, financial concerns might serve as the deciding factor for Chicago even putting a claim in on Ramirez. The 38-year-old is owed approximately $4.5 million, with about $3.3 million deferred, but a FOXSports report Thursday claimed Ramirez wanted a one-year extension to waive his no-trade clause.
White Sox free agents, such as first baseman Paul Konerko and catcher A.J. Pierzynski, could free up some salary room if they depart after this season. The White Sox still seem unlikely to commit to a player past the final month of the 2010 campaign who would primarily serve as the designated hitter.
"Ideally, that's always my preference -- help for today and help for tomorrow as well," said Williams, referring to the claim of Alex Rios last August, aimed to help in 2010 and beyond. "Sometimes you're in the mode of just getting into the playoffs and you shoot for help today. Every name that goes on the wire, we take a look at and explore the possibilities.
"Whenever I've talked to [White Sox chairman] Jerry [Reinsdorf], it's less about money even though we are stretched right now. We are stretched, and quite frankly looking at walkups every day to see exactly what we're going to have available to us. But he certainly hasn't given me any reason to think there are any restrictions right now on dollars.
"You certainly don't want to take on a contract of player X when player X had two or three years left, and you don't think he'll be productive for those two or three years down the line, because what does that do to you? That limits your ability to compete in those years, limits your payroll flexibility, so you have to be very careful and make sure a guy fits into the equation, like I felt Alex did last year."
An announcement will come Friday as to the team receiving the Ramirez claim. The Dodgers and that team then have until Tuesday, the final day for a player to be added to be eligible for the playoff roster, to finish a move. The Dodgers also could let that particular team take Ramirez and his full contract or pull him back.
Any team has a chance to make a deal for Ramirez if he goes through waivers unclaimed. But in a show of full support for the team currently assembled, White Sox manager Ozzie Guillen isn't looking for any late addition to save this playoff cause.
"My names, they're in the lineup. I swear to God," Guillen said. "I don't get into those details about baseball, those details about people. What I've got here, I put it there and I try to make the best of them.
"I've already got 25 problems, I don't want to worry about another one that's not here. Kenny knows what is going on and I think it's very complicated to my knowledge about bringing players on. What I have is in there, and I'm very happy with that. I'm very comfortable with this ballclub. I'm very pleased with what we have."
Scott Merkin is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.