Williams said initial optimism that the Sox could quickly re-sign the slugging free agent have dimmed at baseball's annual winter meetings in Florida, the Chicago Tribune reported.
"At this point we have no choice" but to explore other options, Williams told the newspaper. "All I can say is we've got meetings with other guys' representatives (Tuesday) with the idea in mind (to) get a deal done."
The Tribune reported that Konerko's agent met with team owner Jerry Reinsdorf Tuesday in what was described as a last-ditch effort to get the team's best hitter to accept a new deal, but snags were apparently hit in terms of money and the length of the contract.
Konerko last year led the team in batting average (.312), homers (39) and RBI (111). The White Sox this week signed slugger Adam Dunn, who is slated to be a designated hitter but who also could play first base.