"It is going after a symptom rather than the cause of the problem in the first place," he said.
Cornyn, the head of the GOP Senate campaign committee, softened his stance while visiting a port of entry along the U.S.-Mexico border. He met with the Texas Border Coalition, a group of local mayors and business leaders focused on border security and traffic backlogs at ports that affect commerce.
The coalition also opposes calls to deny citizenship to children born in the United States but whose parents are illegal immigrants. Birthright citizenship is a constitutional right under the 14th Amendment.
Senate Republicans recently have suggested that right may be due for a second look amid the raging national debate over immigration. Supporters argue a change will discourage immigrants who have children on U.S. soil to use them to stay in the country.
Earlier this month, Cornyn was quoted in The Dallas Morning News as saying, "We need to have hearings. We need to consult constitutional scholars and study what the implications are."
But Cornyn said Friday his new remarks didn't amount to his stance shifting.
"I was asked a question whether I would participate in hearings on the issue," Cornyn said. "And I said, 'Well, sure, I'm happy to do my job and listen to expert witnesses.' But upon further reflection I really think it is a symptom and not the cause."
He then took aim at border security.
"I think it would be a moot issue if, in fact, the federal government was doing its job of border security and enforcing the law," Cornyn said.
A Pew Hispanic Center study released this month said that one in 12 children born in the U.S. has at least one parent who is in the country illegally.