No stranger to chasing down his white whales, Ken Williams harpooned his latest one on Thursday.

And it’s a doozy.

The Sun-Times first reported back in June that slugger Adam Dunn was sought by the White Sox general manager, but Washington’s asking price in completing the trade became completely unrealistic.

With Dunn in open water and a free agent this winter, however, opening the checkbook was the only move they had to make.

According to several sources, the Sox finalized a four-year, $56 million deal with Dunn, and were prepared to make the move official Friday afternoon, after he passed his physical.

Williams was his usual coy self when asked about it, especially because he was on a teleconference to talk about the news that the Sox non-tendered closer Bobby Jenks on Thursday afternoon.

“I guess since it’s out there … why don’t we talk about that tomorrow at 2 o’clock at the ballpark,’’ Williams said, referring to a press conference that the Sox will obviously be calling.

It was also learned on Thursday that Dunn is not the lone free agent on the Sox radar, with Paul Konerko also being hunted by Williams.

He was a bit more forthcoming about that topic concerning the Sox team captain and his free agency.

“I’ve made no secrets that we have strong interest in bringing him back,’’ Williams said of Konerko. “Not only is there room for it, but it would be the ideal fit from our perspective. The one thing Paul and I talked about at the end of the season is being mindful of each other’s process. It’s a fine line I have to walk right now where I am respectful of his process, but where we are also putting the best baseball team on the field.’’

Later, Williams was asked about the talks concerning free agent A.J. Pierzynski, and again brought up the pursuit of Konerko as a big reason why Pierzynski is on hold for now.

“We’ve had conversations, we’ve had a lot of conversations,’’ Williams said of bringing his catcher back. “Again, I value A.J.. I would love to have him back. My dilemma right now is do you take such an overture and let it take you out of the payroll zone that you may need to get a Paul Konerko back. We’re in a holding pattern because to be honest the next thing I would like to put back in a Sox uniform is Paul Konerko.’’

The problem is that the Pierzynski ship may have already sailed. A source close to the situation said that Pierzynski was close to signing with Toronto.

That would leave the Sox going with Ramon Castro and Tyler Flowers, or pursuing Miguel Olivo.

One player they will no longer have to pursue is Dunn.

The 31-year-old left-handed hitter ranks second in the majors to only Albert Pojols in homers since 2004 with 282. He is coming off a 2010 season in which he hit 38 homers and 103 RBI for the Nationals. He will likely see a majority of the time as the Sox DH, with Williams insisting that Carlos Quentin will be his starting right fielder again on Thursday.

So where is all the money coming from, with even Alexei Ramirez getting a bump on Thursday when the Sox picked up his $2.75 million option? Williams explained it as they were either going to be the haves or the have nots this season, and with so much already invested in pitching, they had no choice but to be players.

“I talked to [board chairman] Jerry [Reinsdorf] at the end of the year and usually there are four plans I give him,’’ Williams explained. “This year there were two. One was a young team and the other was we were going to add to the mix. We just didn’t want to be in the middle. If you’re going to be all in you go all in.’’

As far as Jenks, Williams wouldn’t close the door on him completely, but not at the price Jenks would have received in arbitration.

“Bobby Jenks brought a World Series to Chicago and I will never forget that,’’ Williams said. “And I haven’t closed the door on that. All it says today is with the dollars it will bring in an arbitration hearing we cannot go there. The message I have to Sox fans and to Bobby Jenks is he gave us everything he had. I’m proud of his growth as a person, a player and as a dad. I wish the best for him if indeed he does end up somewhere else.’’

Williams likes the fact that Matt Thornton, Chris Sale and even Sergio Santos can all close games, and plus next week the winter meetings begin.

More whales for Williams to chase.

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