Greene, who will turn 21 on Wednesday, is currently rehabbing his right elbow after having Tommy John surgery on June 19.
Greene was playing with Class A Hickory of the South Atlantic League, where he was hitting .218 with five home runs and nine RBIs in 25 games. Greene was the Rangers' 16th-round Draft pick in 2013 out of New Mexico Junior College.
He's a career .250/.390/.357 hitter in 577 professional plate appearances.
"Marcus is a guy that can hit home runs, is a great athlete, he has some strength, he has an idea for the strike zone, and just did a great job in a short time with the Rangers," Preller said. "He's a guy that obviously has had a setback this year, and he had Tommy John surgery."
Venable was originally in Tuesday's lineup against the Braves, hitting sixth and playing center field.
The 32-year-old had been the longest-tenured Padres player and could become a free agent after the season, was hitting .258 with six home runs and 30 RBIs in 98 games. He can play all three outfield positions and he bats left-handed and runs well.
"[The trade] was a surprise, of course. But these things happen in baseball," Venable said by phone. "But being realistic about free agency, we had already braced ourselves for the Trade Deadline. We knew it was a possibility."
Venable, who made his big league debut for the Padres on Aug. 29, 2008, was a seventh-round Draft pick in 2007 out of Princeton.
"In leaving, you immediately reflect on things. For me, it's all about the people. That's what I'll miss the most," Venable said.
He's a career .252/.316/.410 hitter and had his best season in 2013, when he hit .268 with 22 home runs and 53 RBIs.
"We felt it was a deal that makes us better in the future," Preller said.