Donald Trump’s interview Tuesday with Rep. Michael T. McCaul for secretary of homeland security set off alarms for illegal immigration opponents who view the congressman as weak on border security and latently pro-amnesty.

Mr. McCaul, Texas Republican and chairman of the House Homeland Security Committee, has repeatedly riled conservatives with what they describe as ineffective posturing against President Obama’s lax enforcement of immigration laws.

His chief offense was co-authoring the 2015 Secure Our Borders First Act, which conservatives and activists reviled as a fig leaf to Democrats that included tough language but ignored the estimated 11 million illegal immigrants living in the U.S. and kept too much decision-making in the hands of the Obama administration.

“We certainly hope that Donald Trump would not reward a deceptive pro-amnesty lawmaker like Michael McCaul with a Cabinet position,” said William Gheen, president of Americans for Legal Immigration PAC. “That would be very disappointing to all of us that believed his campaign promises to secure our borders and deport millions of illegal immigrants under current U.S. laws.”

He noted that Mr. McCaul also was among 19 House Republicans who in 2014 signed onto House Speaker John A. Boehner’s “Principles on Immigration Reform” agenda, which included a path to citizenship and voting rights for illegal immigrants.

The homeland security secretary will be one of the most prominent nominations for Mr. Trump, who made fighting illegal immigration the cornerstone of his campaign. He promised to implement aggressive deportation policies and build a wall along the U.S. border with Mexico.

That’s why Mr. McCaul would be a surprising choice, said Jessica M. Vaughan, director of policy studies at the Center for Immigration Studies, which supports a crackdown on illegal immigration.

“I don’t doubt Mr. McCaul’s commitment to homeland security, but I do wonder about his commitment to controlling immigration,” she said. “He has been quite moderate and conventional on border security issues, eschewing any bold moves to address the problems created by the Obama administration’s policies, for example.”

Mr. McCaul’s border security bill, she said, exposed the chairman’s weak stance on the issue while unaccompanied children were surging into the United States.

“Even in the midst of a crisis at the border, when tens of thousands of new arrivals from Central America and other nations were streaming across to be released under Obama policies, he introduced a border security bill that avoided the critical issues that were fueling the crisis,” said Ms. Vaughan. “This bill was panned by the most prominent Border Patrol associations, even though it would have been a huge infusion of funding, because it ignored the key problems such as the catch-and-release policy.”

Mr. McCaul first wrote his border security bill in 2013, working with Democrats on a compromise that demanded the Homeland Security Department come up with specific plans for curtailing illegal crossings.

Meanwhile, Senate Republicans were advancing a proposal that called for specifics: 20,000 more Border Patrol agents, hundreds of miles of additional fencing and clear-cut technology goals.

Mr. McCaul’s office didn’t respond to a request for comment Tuesday, but he has defended his approach in the past by saying he didn’t want to dole out resources without having a strategy in place.

House Democrats liked Mr. McCaul’s approach and even adopted it as their own, using it as the border security component of a broad immigration overhaul bill in the previous Congress.

After his meeting with the president-elect, Mr. McCaul told reporters at Trump Tower that the conversation was “substantive and productive.”

In response to reporters’ questions, Mr. McCaul described border security in terms of fighting terrorism.

“I articulated to the president how we need to close off all terror pathways into the United States. We need to secure our borders and secure the United States from these terrorists coming into the United States and perpetrating acts of terror like what we saw yesterday,” he said, referring to the attack at Ohio State University.

Trump transition team officials have extolled the president-elect for bringing a diverse group of candidates into the interview process.

Still, Mr. McCaul isn’t the only candidate to provoke criticism.

Mr. Trump’s supporters also balked at his consideration of Mitt Romney for secretary of state, saying the 2012 Republican presidential nominee was an unforgivable foe who tried to undermine Mr. Trump’s campaign and represented the worst tendencies of the party’s establishment.

Activists against illegal immigration were quick to name someone they deemed better suited than Mr. McCaul for homeland security secretary.

Rosemary Jenks, director of government relations at NumbersUSA, which lobbies for stricter immigration enforcement, said the Homeland Security Department needs a leader with knowledge of law enforcement and immigration policy.

Her favorite for the job was Kansas Secretary of State Kris Kobach, who also serves as legal counsel to the Immigration Law Reform Institute and worked as counsel to U.S. Attorney General John Ashcroft. Mr. Kobach is advising the Trump transition team and has met with the president-elect.

“That immigration component is absolutely critical if Donald Trump wants to move his agenda forward in a realistic and effective way,” she said. “Mike McCaul is not in any way, shape or form that we have seen evidence of an expert on immigration policy.”

Other candidates interviewed for homeland security secretary include Maricopa County (Arizona) Sheriff Joe Arpaio, Milwaukee County (Wisconsin) Sheriff David A. Clarke Jr., former New York Mayor Rudolph W. Giuliani and retired Marine Gen. John F. Kelly.

Mr. Gheen said the Americans for Legal Immigration PAC was backing Sheriff Arpaio.

Sheriff Arpaio, known as “America’s toughest sheriff,” has been on the forefront of the fight against illegal immigration. He lost a re-election bid Nov. 8 after 24 years in office.

“We would love to see Sheriff Joe Arpaio put in charge of the Department of Homeland Security or Border Patrol,” Mr. Gheen said. “We put that idea online last week and received a huge positive response.”

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