"What Mr. Trump is offering is not conservatism, it is Trump-ism -- a toxic mix of demagoguery and nonsense," the former Texas governor said in a statement.
Perry, who has also criticized President Barack Obama's efforts on securing the border with Mexico, said Trump "believes states should fend for themselves on border security."
"Not only is this wrong, it perpetuates the same failed policies that have left our southern border porous and vulnerable," Perry said.
The next president must enforce the federal government's obligation to secure the border, Perry added.
"America doesn't need another president who will pass the buck on border security," he said. "We need a president who will finally act to secure the border after decades of failed leadership in Washington, D.C. And Mr. Trump has done nothing to prove that he is the president America needs."
In a Twitter response on Thursday, Trump not only criticized Perry's ability to secure the border, but his intelligence, as well.
"@GovernorPerry failed on the border. He should be forced to take an IQ test before being allowed to enter the GOP debate, Trump tweeted.
Perry's comments are among those of multiple Republican presidential candidates
who have criticized the real estate mogul's rhetoric about undocumented immigrants and his plan to build a "great, great wall on our southern border."
Perry has looked to parlay his executive experience from 14 years as governor to distinguish himself from his Republican rivals. As a border state governor, he's also highlighted his own record boosting the presence of patrolmen in the Rio Grande Valley.
Speaking to CNN's Wolf Blitzer on Wednesday, Perry said he was the best candidate in the 2016 race on border security.
"I know how to do this, Wolf. You put the personnel on the ground, you have the aviation assets ... looking down 24/7 with quick response teams to those activities that are either clearly illegal or suspicious. That's how you secure the border. We know how to do this," Perry said.
Last weekend, Trump attracted thousands at a rally
in Phoenix, where he again criticized U.S. efforts to prevent illegal immigration and negotiate trade deals favorable to the U.S.
The event was slammed by Arizona Republican Sen. John McCain.
"This performance with our friend out in Phoenix is very hurtful to me," McCain said in a recent interview with The New Yorker
. "Because what he did was he fired up the crazies."
McCain also told the magazine that Trump's immigration policy is "very bad" and erodes the progress Republicans have made on immigration.
Trump tweeted Thursday that McCain should apologize for calling the "amazing Americans" who attended the Phoenix event "crazies."
".@SenJohnMcCain should be defeated in the primaries. Graduated last in his class at Annapolis--dummy," he said.