The chief of the U.S. Border Patrol is suggesting "jail time" and tougher "consequences" to help America crack down on the number of migrants crossing illegally over the U.S.-Mexico border. 

Jason Owens made the remark Thursday to CBS News as a large group of illegal immigrants were captured on video bursting through razor wire and surging towards a border wall in El Paso, Texas. 

"I'm talking about jail time. I'm talking about being removed from the country, and I'm talking about being banned from being able to come back, because you chose to come in the illegal way instead of the established lawful pathways that we set for you," Owens said. 

Owens added, "I think we need to take a look at the asylum laws and make it where only people that have a legitimate claim can claim asylum." 

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"If there's no motivation to do it the right way -- and the right way, it is causing people to have to wait a little bit longer -- well naturally, they're going to choose to come between the ports of entry," Owens also told CBS News. "We need to take that off the table and make sure everybody's coming through the front door." 

This week, the Supreme Court briefly allowed Texas’ anti-illegal immigration law, which allows police to arrest illegal immigrants, to go into effect despite a legal challenge from the Biden administration. The case was kicked back down to the U.S. Court of Appeals for the Fifth Circuit, which then blocked the law again hours later as arguments proceed on the merits. 

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The Biden administration has claimed that the law encroaches upon federal responsibility over immigration enforcement.  

"S.B. 4 will not only make communities in Texas less safe, it will also burden law enforcement, and sow chaos and confusion at our southern border. S.B. 4 is just another example of Republican officials politicizing the border while blocking real solutions, White House press secretary Karine Jean-Pierre said in a statement this week. 

But Owens told CBS News on Thursday that the law is "not going to stop us from doing our job," and there is "no better partner for the Border Patrol than the Texas Department of Public Safety." 

"We have worked hand in hand with that agency for as long as I've been around, and I don't see that ever stopping," he added. "They have always been very good at complementing our mission. They back us up when we're out in the field, and we do for them as well. So, whatever the laws are that they're going to be enforcing, our mission remains constant. Their mission remains constant." 

Fox News’ Adam Shaw, Bill Melugin and Michael Lee contributed to this report.