“At any moment, FedEx can tell you where that package is. It’s on the truck. It’s at the station. It’s on the airplane,” Mr. Christie told the crowd in Laconia, N.H. “Yet we let people come to this country with visas, and the minute they come in, we lose track of them.”

He added: “We need to have a system that tracks you from the moment you come in.”

He said 40 percent of illegal immigrants are allowed into the United States legally with a visa and then stay longer than their visa allows.

“However long your visa is, then we go get you,” Mr. Christie said. “We tap you on the shoulder and say, ‘Excuse me. Thanks for coming. Time to go.’ ”

Mr. Christie has been lagging in recent opinion polls and is in danger of not being one of the top 10 candidates who will be invited to participate in the next official Republican debate, on Sept. 16.

With Mr. Trump taking a hard line on illegal immigration, other Republican presidential candidates have sought to toughen their stances as well.