In a sometimes-testy exchange during last night’s Republican debate, Mike Huckabee and Mitt Romney argued over whether illegal immigrants should be allowed to receive state-based merit scholarships.

When he was governor of Arkansas, Mr. Huckabee supported a bill that would have made some illegal immigrants who grew up attending schools in the state eligible for a merit-based aid program, known as the Arkansas Academic Challenge Scholarship. (The legislation passed the state’s House of Representatives but died in the Senate.)

A woman from Texas had asked Mr. Huckabee about his stance on the topic.

The former governor replied that he believed it was appropriate to give the children of illegal immigrants who had “been in our schools their entire school life the opportunity to have the same scholarship that their peers had that had also gone to high school with them and sat in the same classrooms.”

He noted that the bill would have required those students to be in the process of applying for U.S. citizenship before they could receive the scholarship.

“We’re not going to punish a child because a parent committed a crime,” he added. “That’s not what we typically do in this country.”

Mitt Romney, a former governor of Massachusetts, was prompted to respond by the debate’s moderator, Anderson Cooper of CNN.

Mr. Romney said that by supporting scholarships for illegal immigrants Mr. Huckabee was essentially in favor of giving away “taxpayer money” to people who had broken the law. Doing so, he said, would limit the amount of state funds available for scholarships to help other, legal residents.

“The right thing here is to say to people who are here legally as citizens or legal aliens, we’re going to help you,” Mr. Romney said. “But if you’re here illegally, you ought to be able to return home or get in line with everybody else.”