A number of players that the Rangers have been rumored to potentially have interest in this winter have been given a qualifying offer by their former team to preserve compensation. I'll update this list as more information is known.
B.J. Upton has obtained a qualifying offer from the Tampa Bay Rays. The Rangers have been rumored to be interested in Upton to potentially replace Josh Hamilton in center field.
Michael Bourn has obtained a qualifying offer from the Atlanta Braves. Like Upton, the Rangers could look to Bourn to replace Hamilton in the outfield. Unlike Upton, however, Bourn would be more of a top of the order hitter and perhaps someone who could replace Ian Kinsler as the leadoff hitter.
Adam LaRoche has obtained a qualifying offer from the Washington Nationals. While LaRoche hasn't been mentioned in many rumors as a potential Rangers' target, he could fill a need at first should the Rangers decide that Mitch Moreland isn't a long term option.
Here are players who were not given a qualifying offer that the Rangers may be interested in:
The Yankees have declined tendering Russell Martin a qualifying offer. Martin will likely interest the Rangers with their gaping void at catcher now that Mike Napoli will be free to sign with any team he likes. Perhaps the Rangers roll with a tandem of Martin (or someone like Martin) and Geovany Soto and use the money saved on what it would have cost to sign Napoli on options that they feel would better improve the team.
The Anaheim Angels declined tendering Torii Hunter a qualifying offer. Hunter is another possible Hamilton-bat/outfielder replacement. While Hunter is my least favorite baseball player, he had quite a season for the Angels last year (5.3 fWAR, his best season of his career and his first season above 4 fWAR since 2002), and is likely someone who will be on the Rangers' radar. Now there is the added bonus in that he won't cost the Rangers a draft pick to sign.
Hamilton being offered the qualifying one-year contract was a much easier decision to make for the Rangers than the Mike Napoli offer decision simply because there's nearly zero chance that Hamilton accepts the $13.3MM offer, and should he, the Rangers would be thrilled. Hamilton has until next Friday to accept or decline the one-year offer. Afterward, when he declines, should Hamilton sign elsewhere as expected, the Rangers would receive a compensation draft pick.