While the Astros have seemingly fallen out of the mix because of financial limitations, the Rays have emerged as the latest club to show interest in Soriano, who on Monday night put himself in position to be traded when he opted to accept arbitration.
There's at least a sense that Soriano could be traded before this year's Winter Meetings conclude Thursday. But instead of setting a definitive timetable, Wren is simply providing further indication that he's confident that a deal could be completed in the very near future.
"When you're making deals, you don't know exactly how close another team is to pulling the trigger to say 'Yea' or 'Nay,'" Wren said. "We know who has interest. We know who has the most interest and we've started down some of those paths."
While some media outlets chose to define the Rays as the "front-runners" for Soriano, the Braves contend that they are still in discussions with a handful of teams. Wren said that he met with six clubs that are interested in the right-handed reliever Wednesday.
Tampa Bay could benefit from adding a power arm to the back end of its bullpen mix. But there is some reason to wonder whether the Rays want to pay the $6.5 million-$7 million that Soriano will gain courtesy of the one-year non-guaranteed contract that he gained by accepting arbitration.
Wren confirmed that he hasn't discussed the possibility of picking up a portion of this salary that Soriano could be awarded through the arbitration process.
The Mets, Phillies, Yankees, Orioles and Angels appear to be the other clubs that have shown interest in Soriano. The Braves obviously could prove somewhat reluctant to move him to a division rival (Phillies or Mets).
"There are some clubs that are more interested than others, and there are clubs that we match up with better than others," said Wren, who has spent most of his time this week attempting to utilize his surplus of arms to work a deal with pitching-hungry teams.
With the Soriano discussions currently drawing a majority of his attention, Wren will patiently wait to attempt to move one of his high-priced starting pitchers.
Once a market is established for John Lackey and other top free-agent starting pitchers, Wren will gain more clarity regarding his ability to deal Derek Lowe. At this time, he'll also gain a better sense about whether he will eventually have to resort to Plan B -- which would be to trade Javier Vazquez.
Wren's current priority is to continue working on these trades involving his pitchers. But at the same time, he is continuing to search for a player who could satisfy his offensive needs by primarily playing first base and occasionally playing one of the corner outfield positions.
Free-agent outfielder Xavier Nady, who appears to be near the top of this wish list, recently increased his rehab program and remains confident that he'll be fully recovered from Tommy John elbow ligament replacement surgery by the start of Spring Training.
Mark DeRosa stands as another player who would be versatile enough to play both first base and the outfield. Still, while some of Atlanta's players have lobbied for his return to the organization, it's clear that their wishes won't be fulfilled unless DeRosa's salary demands decrease to the neighborhood of $4 million-$5 million.
"There are people we're talking to all of the time," Wren said. "We're expressing interest in the guys we targeted back in October. Some of them we engage in, and some of them, we don't. Some of them we stay engaged until the end, and some of them, we don't."
While Wren has said he'll likely fill his offensive needs via free agency, there may be some interest in Nick Swisher to fill this first-base/outfield role if the Yankees deem him expendable now that they have added Curtis Granderson to their outfield mix.
The Braves had interest in Swisher before the White Sox dealt him to the Yankees last year. But for now, it appears they may have more interest in Nady, who has told friends and associates that he would welcome the opportunity to play in Atlanta.
While Soriano's arbitration decision has thus far proven to be the most newsworthy event experienced by the Braves this week, Wren said that he has been busy while conducting 20 face-to-face discussions with clubs regarding trades. During last year's slow-paced Winter Meetings, he had just one such formal meeting with another club.
Mark Bowman is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.