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Owens had produced scoreless outings his first two times out, but he has given up eight runs over 5 1/3 innings in his past two starts.
"Because of today and the last two outings, that's not to say he's completely eliminated," Red Sox manager John Farrell said. "But we're also at the midpoint in games in the schedule, so the competition starts to clarify itself the deeper you go."
Steven Wright and Roenis Elias are probably ahead of Owens in the race for the final spot at this point. Boston has a rotation spot available only because Eduardo Rodriguez will start the season on the disabled list with a right knee injury.
Owens has tried to put the race out of his mind and focus on the things he can control.
"I'm trying to get outs," Owens said. "I'm not trying to fall into any mind traps out there. It's a mental game enough. I'm trying to get outs and prepare myself for the season, whatever position that's in. I just want to help this team."
What hindered Owens on Thursday?
"The third inning, I felt like [there were] some mishaps with my mechanics," Owens said. "I was thinking about it too much on the mound. I had a few conversations in the dugout about what [catcher Ryan] Hanigan saw. I'll work on that tomorrow and continue to work on it."
"I think he just needs to tighten up his fastball command a little bit," Hanigan said. "He threw some pretty good changeups. He threw some decent curveballs. But overall, his fastball command is a little erratic."
The big hit Owens gave up was a three-run homer to Christian Walker, which came after back-to-back walks.
"Yeah, the command seemed to come and go a little bit today," Farrell said. "Home runs are going to be allowed, but it's probably the couple of walks leading up to the home run that you'd rather not see."
Owens, 23, made 11 starts for the Red Sox in 2015, going 4-4 with a 4.57 ERA. Knuckleballer Wright was 3-4 with a 3.96 ERA in nine starts and also has the versatility to pitch out of the bullpen. He is out of Minor League options.
Lefty Elias, acquired in a December trade with the Mariners, has made 49 starts over the past two seasons, going 15-20 with a 3.99 ERA. Elias and Owens both have options left.