"We've got to make room and got to do what's necessary for the team," Claiborne said. "That's what I'm about, and I'm just going to go down there, get my work in, and when they need me, they'll call me."
Claiborne, 25, was 0-1 with a 2.88 ERA in 34 appearances for the Yankees, finishing 10 games. In 40 2/3 innings, he allowed 13 earned runs and 37 hits with 11 walks (four intentional) and 33 strikeouts.
Manager Joe Girardi said that the Yankees felt they needed to have arms capable of offering more distance like Adam Warren and David Huff, with long games always in play at Fenway Park as well as a day-night doubleheader next Tuesday against the Blue Jays at Yankee Stadium.
"Going into what we have, the games here and the doubleheader, we felt like distance was probably a little bit more [important]," Girardi said. "Clay's pitched really well. I'm very pleased with what he's done, but one of those long guys could end up being a starter sometime during that series, so we kept it that way."
Claiborne said that he is hopeful that he has shown enough to be part of the Yankees' bullpen picture later this year, as well as in 2014.
"I want to be a part of this going forward," Claiborne said. "When that time comes -- obviously nobody knows what the future holds, and no position is set. When it comes time that one of those positions comes open, those positions will be fought for next year. This year, it's just a matter of getting back in the hunt and getting back in the playoffs."
Bryan Hoch is a reporter for MLB.com. Follow him on Twitter @bryanhoch and read his MLBlog, Bombers Beat. This story was not subject to the approval of Major League Baseball or its clubs.