Conservative immigration hawks are pushing President Trump to end a program that is shielding thousands of undocumented immigrants from deportation.

Trump on Wednesday signed two executive actions on immigration, but neither addressed Deferred Action for Childhood Arrivals (DACA), a program started under President Barack Obama Barack Hussein ObamaThe Hill's 12:30 Report - Presented by Facebook - Don't expect a government check anytime soon Trump appointees stymie recommendations to boost minority voting: report Obama's first presidential memoir, 'A Promised Land,' set for November release MORE for people brought to the United States illegally as children.

The Trump administration remains mum on its plans for the program, which reportedly is still accepting new enrollees.

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“These are work permits to illegal immigrants who didn’t have them before. That can’t be allowed to continue,” said Mark Krikorian, who leads the Center for Immigration Studies.

He added that if the Trump White House doesn’t end DACA soon, “it makes the program not just Obama’s illegal action, but one that the Trump administration has at least acceded to.”

Trump took a tough line from the start of his campaign about Obama’s immigration actions, DACA and the related Deferred Action for Parents of Americans.

“[Obama’s] horrible order that he signed on illegal immigration, where people are just flowing into the country, horribly, without any checks or balances — we don’t even know who’s coming in. That would be ended immediately,” Trump said in an interview with Fox News’s Sean Hannity about a year before his general election victory.

“We will immediately terminate President Obama’s two illegal executive amnesties in which he defied federal law and the constitution to give amnesty to approximately 5 million illegal immigrants,” Trump said at an August speech in Arizona, where immigration hard-liner Joe Arpaio introduced him.

But Trump has at other times appeared to advocate a softer approach.

In December, he said that he is “going to work something out” for people in the program.

“They got brought here at a very young age, they’ve worked here, they’ve gone to school here. Some were good students. Some have wonderful jobs. And they’re in never-never land because they don’t know what’s going to happen,” Trump said in an interview with Time.

In an interview that aired Wednesday, Trump said DACA participants shouldn't be "very worried."

“They shouldn’t be very worried, I do have a big heart. We’re going to take care of everybody, we’re going to have a very strong border, we’re going to have a very solid border,” Trump told ABC's David Muir. “You have great people that are here that have done a good job, they should be far less worried.”

In Congress, meanwhile, there is support in both parties for allowing people brought to the U.S. as children to stay.

During a press briefing Wednesday, White House press secretary Sean Spicer said that Trump’s new immigration orders don’t “specifically deal” with DACA. He said that Trump and his team would reach a deal with the program in a “humane way.”

Earlier in the week, Spicer reiterated that Trump’s priority is removing undocumented immigrants who have committed crimes. Trump plans to deal with “the entire number of folks that are here illegally” after that, he added.

The open-ended language is stirring alarm among immigration hawks, who are looking to Trump to enact long-awaited policy changes.

NumbersUSA, a group that supports lowering immigration levels, called on its members to launch a social media campaign demanding that Trump follow through on his campaign promise.

While the email said the group would give Trump the benefit of the doubt, “it has become clear the last two days that Pres. Trump will not end the renewals and issuance of DACA work permits ‘immediately’ as he stated repeatedly in the campaign.”

Krikorian said he hasn’t heard back from the White House about the status of DACA, but said it’s possible that the administration might be holding out for more leverage in a future negotiation with Congress.

He framed the delay as a possible victory by the more establishment voices in Trump’s White House, represented by chief of staff Reince Priebus, over the more populist ones, embodied by chief strategist Stephen Bannon.

“My concern is whether this suggests that there is too much flexibility in this administration in giving in to establishment concerns,” he said.

“The subset of that fear is that they will give away DACA for too little because the Republican establishment would be happy to pass a phony-baloney border bill.”

But other Trump supporters say they are heartened by the immigration actions he has taken so far.

Conservative commentator Ann Coulter pointed to the text of one of Trump’s Wednesday orders, which directs executive branch agents to “detain individuals apprehended on suspicion of violating federal or state law, including federal immigration law.”

“I don’t see how this doesn’t fully contravene DACA,” she said.

And James Carafano, the Heritage Foundation’s vice president who ran the Department of Homeland Security transition team for Trump but has not spoken to the White House about the issue since the inauguration, told The Hill that Wednesday’s actions are a signal that the White House is prioritizing the issue.

“Today came, for us who would like to see delivery on this, as something to be excited about … that we are going to enforce the law,” he said.

But he added that “we are all asking the same question, which is what’s he going to do about [DACA]?”