"We realize [his return is] going to be a year down the road, and I think [the injury] caused other people not to take him," Meek said. "But we feel pretty confident in the Tommy John surgery process. It'll be a year late, but we just felt like where we had him on our board at that spot we couldn't pass him up.
"We're willing to take a chance. We think we can sign him at a very reduced cost, and that makes him attractive to us down in that spot."
The Draft concludes on Saturday, with exclusive coverage of Rounds 11-40 beginning on MLB.com at 1 p.m. ET.
Taken 287th overall, the 5-foot-11, 212-pound lefty had three solid seasons as a Longhorn. Peters has a curveball that ranges from 76-80 mph, a sinking fastball and a deceptive circle changeup. He ended his junior season at 7-3 with a 2.13 ERA in 80 1/3 innings.
Peters combined for a shutout in his last turn on the mound, throwing five four-hit innings in No. 5 seed Texas' upset of top-seed Oklahoma State in the Big 12 Championship. He hadn't given up a run in his previous three starts.
Although he doesn't have the build of a classic big league starter, scouts like Peters because he pitches to contact and can locate on both sides of the plate as well as down in the strike zone. He struck out 53 batters and held hitters to a .247 batting average.
Peters' team began play in the Austin Super Regional on Friday with a 4-2 victory against the University of Houston and will continue its postseason journey without him.
Maria Torres is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.