According to Rosenthal, sources indicated that the Boston Red Sox, Chicago Cubs, Baltimore Orioles, Cleveland Indians, Colorado Rockies, Chicago White Sox, Detroit Tigers, Oakland Athletics, Pittsburgh Pirates and Toronto Blue Jays are on his no-trade list.
While most of the teams on Darvish's list are either not in contention or not in the market for a rental starting pitcher, it could come into play.
The Cubs have been linked to Detroit Tigers starting pitcher Justin Verlander, per MLB.com's Jon Morosi, while ESPN's Buster Olney listed the Indians as a team that could have interest in Oakland Athletics ace Sonny Gray.
ESPN's Jerry Crasnick reported this week that Rangers general manager Jon Daniels began calling teams regarding Darvish and that he is ready to move the Japanese righty for the "right deal."
Darvish is set to become a free agent during the offseason, but he has all the makings of an ideal hired gun who can join a contending team and help lead them deep into the playoffs.
Among the teams in the market for a quality starter are the New York Yankees, Los Angeles Dodgers, Washington Nationals and Houston Astros.
So far this season, Darvish is 6-8 with a 3.44 ERA, 1.12 WHIP and 143 strikeouts in 133.1 innings.
The 30-year-old veteran is a four-time All-Star, and he boasts a career MLB ERA of 3.32 with a career strikeout rate of 11.0 per nine innings.