According to David Kaplan there are two AL teams in the mix for Dexter Fowler.
The White Sox are in play for Fowler as are the Texas Rangers. He is hearing from a few teams that still want a bat.— David Kaplan (@thekapman) February 1, 2016
Fowler has been one of the forgotten outfielders this offseason, as there was an abundance of talent available. While he's not as valuable as someone like Jason Heyward, or Alex Gordon, over the course of Fowler's major league career, he's proven to be an above average hitter.
His career OBP is an excellent .363, and Fowler would become a top-of-the-lineup threat for any team that eventually signs him. Of the two teams that Kaplan mentioned, the White Sox likely need him more than the Rangers.
If the season began today, Chicago's 2016 outfield, from left to right, would be Melky Cabrera, Adam Eaton, and Avisail Garcia. After a slow start in 2015, Eaton turned his season around by hitting 14 home runs (a career high), and posted a slash line of .287/.361/.431 with a wOBA of .346, and a wRC+ of 118. However for Cabrera, and Garcia, neither player proved to have much value.
Together, they combined for an fWAR of -1.4, and there's little reason to expect those figures to change. Neither one is a decent defensive outfielder, and over the course of their respective MLB careers, they've been around average on the offensive side.
As for the Rangers, their starting outfield is currently Josh Hamilton in left, Delino DeShields in center, and Shin-Soo Choo in right. While that's undoubtedly better than what the White Sox are set to trot-out, it's not by much; especially considering Hamilton's injury troubles, and Deshields' lack of big league experience. Assuming that Hamilton misses significant time in 2016, the Rangers would have Joey Gallo or Ryan Rua set for back-up duties.
While Gallo is an elite prospect, he had a difficult 2015, and could likely use more time in the minor leagues. Rua only has 195 MLB plate appearances to his name, and he hasn't proved to be a valuable asset. Texas could theoretically turn to Nomar Mazara, but he's only 20 years old, and has just 88 plate appearances above AA.
It's unclear how much Fowler is asking for at this point, but both teams should seriously consider signing him. They each have holes in their outfield, and while it's unclear how much money Fowler is asking for at this point, it likely won't be a franchise altering move.