The Rangers have not commented on the reports.
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The 24-year-old is 8-2 with a 3.34 ERA in 17 starts for the Phils this season, striking out 103 batters in 91 2/3 innings. Velasquez is under team control through the 2021 season and was one of several pieces acquired by Philadelphia from Houston this offseason in the Ken Giles trade.
The Rangers and Phillies are familiar with each other, as they pulled off a blockbuster before last season's deadline. Texas acquired left-hander Cole Hamels, Jake Diekman and cash considerations from Philadelphia for Jorge Alfaro, Jake Thompson, Nick Williams, Alec Asher, Jerad Eickhoff and Matt Harrison.
Volquez, 33, is 8-8 with a 4.56 ERA in 21 starts this season, but he has a 2.84 ERA in his last five starts. He has a $10 million mutual option for next season or a $3 million buyout.
Volquez began his career with Texas before being traded to Cincinnati in 2007 for Josh Hamilton. He made his lone All-Star appearance the following season and has a 4.32 ERA in 12 years with the Rangers, Reds, Padres, Dodgers, Pirates and Royals.
The Rangers have reportedly shown interest in several young controllable pitchers, including the White Sox Chris Sale and the Rays' trio of Chris Archer, Matt Moore and Jake Odorizzi. Although they'd be open to a rental, Rangers general manager Jon Daniels said they'd prefer a longer-term option.
"I think the only real change is I was hoping we wouldn't consider a rental," Daniels said Sunday. "In my mind, we are open to longterm controllable acquisitions like Cole [Hamels] and down the line. It's still not my preference, but I'd be open to it."
Cash Kruth is a reporter for MLB.com. Follow him on Twitter at @cashkruth. This story was not subject to the approval of Major League Baseball or its clubs.