Borkowski was designated for assignment, which clears a roster spot for Rodriguez, who was activated from the disabled list.
"I can't say I'm surprised," Borkowski said. "I didn't pitch as good as I'm capable of, they had to make a move and I guess looking at my numbers, I was the logical guy."
Borkowski compiled a 6.86 ERA over 18 relief appearances this season. He struggled through May, giving up 10 earned runs over 8 1/3 innings. He yielded six runs in his last outing on Sunday, during a loss to Philadelphia.
"Dave pitched some big innings for us last year and the year before," Cooper said. "He played a lot of different roles for us last year and did a good job. This year, he was just not consistent enough. We put him in some key spots and he didn't deliver."
The Astros have 10 days to trade, release or reassign Borkowski to the Minor Leagues. Because he has a prior outright and has more than three years of big league service time, Borkowski can refuse a Minor League assignment should he clear waivers and become a free agent.
Ideally, Borkowski would like to be traded to a team that has room for him on its big league roster. If not, Borkowski said he'd take the assignment and pitch for Triple-A Round Rock.
"Hopefully, I'll get back on track and hopefully they haven't given up on me completely and they use me sometime later in the season," Borkowski said.
"We'd like to keep him in the system," said Cooper. "It's up to him and 29 other clubs. If he doesn't stay with us he'll find a job someplace. He has a good arm and, for the most part, he throws strikes."
Alyson Footer is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.