Appearing on “Fox News Sunday,” Rep. Mike McCaul (R-Texas) also said that Thad Allen, a retired commandant of the Coast Guard, and Bill Bratton, who formerly led the police departments in Boston, Los Angeles and New York, could gain the trust of House Republicans.

“I think any three of those coming in would gain a lot of respect from the House Republicans and could help us at a time when we're really trying to push this border security measure out of the House of Representatives,” McCaul said.

Sen. Chuck SchumerCharles (Chuck) Ellis SchumerCan Mueller be more honest than his colleagues? Throwing some cold water on all of the Korean summit optimism House Republicans push Mulvaney, Trump to rescind Gateway funds MORE (D-N.Y.) recommended Kelly for the position on Friday.

Napolitano, the current Homeland secretary, announced that she was leaving on Friday to become president of the University of California.

McCaul said that he and Napolitano had several disagreements during her tenure, including border security and what he termed a hesitance to call out “radical Islam” by name.

“I think there's an opportunity here to get a real leader in who can work with House Republicans on security issues, on enforcement issues and someone who will garner the respect of the Congress, which I think right now there's a real problem with that,” McCaul said.