"He wants to start, he has no bones about that," Levine said. "We have had internal conversations about that, but we prefer to keep the exact nature of those conversations private."
Oswalt, who signed with the Rangers as a free agent on May 29, made six starts before he was moved to the bullpen. He has pitched twice in relief, and each time he pitched two scoreless innings.
He pitched the seventh and eighth innings against the Royals on Sunday in a 6-6 game. Manager Ron Washington asked Oswalt if he could keep going and pitch the ninth. Oswalt declined.
"To be clear, I asked him if he could go back out there," Washington said. "He said he was through. So we went to Robbie Ross. Managers do that all the time.
"Since I've been here, when a player tells me he can't go, that's it. It ends right there. For me, it ends right there."
Oswalt declined to comment when approached in the visitors' clubhouse at Fenway Park on Monday. Oswalt is in his 12th season in the Majors and has made just 15 career relief appearances as opposed to 332 starts. He was a two-time 20-game winner with the Astros in 2004-05 and has a career record of 163-95 with a 3.26 ERA.
The Rangers signed him in May after Neftali Feliz went on the disabled list with a sprained right elbow. (Feliz is now out for the season after undergoing Tommy John surgery.) The Rangers signed Oswalt with the understanding -- but not a guarantee -- that he would be used as a starter. That was the case when he was called up on June 22, and Oswalt went 3-2 with a 6.49 ERA in six starts.
The Rangers then acquired Ryan Dempster from the Cubs at the Trade Deadline, forcing Texas to decide between Oswalt and Scott Feldman for the final spot in the rotation. The Rangers gave it to Feldman, who is currently riding a six-game winning streak. So Oswalt went to the bullpen, and that's where he will be used at this point.
"He's part of the ballclub and an important member of the Rangers' staff," Levine said. "We don't expect that to change. We're in the midst of a pennant race and every game counts. It's our job to put together the best possible team we can so we can be competitive day in and day out.
"Alexi Ogando was an All-Star as a starter and went to the bullpen this year, because we felt that's what was best for the ballclub. We always ask our players to be a part of a winning dynamic, and we try to employ our players in the best possible spot. That's what we're doing now."