WASHINGTON — President Trump’s sympathetic remarks about the young undocumented immigrants known as Dreamers — “these incredible kids,” he has called them — were a surprising turn for a man who had vowed during the campaign to “immediately terminate” their protections from deportation.

But they are unlikely to be the last word. Mr. Trump has not ruled out ending the Obama-era program that shields the young immigrants, who have taken little comfort in his comments. And the president is already coming under intense pressure from the immigration hard-liners in his Republican base to keep his promise.

The problem that Mr. Trump faces as he worries aloud about how to handle the young immigrants, who were brought illegally to this country as small children, encapsulates the beating heart of the difficult choices confronting him. In theory, it is a question of laws and numbers, but in practice it is an emotional and often gut-wrenching matter of human lives affected and families at risk.

It also captures the rifts within the White House, where Stephen K. Bannon, the president’s chief strategist, and Stephen Miller, his policy director, are driving a get-tough immigration policy while Reince Priebus, his chief of staff, has counseled a gentler approach.

“To me, it’s one of the most difficult subjects I have, because you have these incredible kids,” Mr. Trump said at a recent White House news conference. He said he would deal with the matter with “great heart,” but nodded to the political difficulty of doing so.

Mr. Trump added: “I have to deal with a lot of politicians, don’t forget. And I have to convince them that what I’m saying is right.”

Sean Spicer, the White House press secretary, said Mr. Trump’s inaction on the issue was rooted in sympathy for the Dreamers, coupled with a desire to create a careful plan for addressing their status. The administration, consumed with sweeping directives to increase deportations of other undocumented immigrants, does not have the capacity to manage the issue, he said.

“His thinking is: ‘We don’t have to deal with this right now. I want to hear more. I want to plan how we deal with this,’” Mr. Spicer said. “Let’s not start to create a problem that doesn’t exist right now. We’re not capable of handling it right now — we’ve got to deal with the first million.”

Inside the White House, the president’s advisers are concerned that he has repeatedly referred to Dreamers, many of whom are in their 20s and 30s, in such sympathetic and politically loaded terms. “Our immigration folks are like, ‘Stop calling them kids,’” Mr. Spicer said.

The president is weighing a variety of strategies for dealing with the roughly 840,000 Dreamers, according to senior officials, including Mr. Priebus, and lawmakers of both parties in Congress have been trying to devise legislation to carve out a special status for them. For the time being, Mr. Trump’s administration is still issuing work permits to undocumented people under the program, leaving their protection intact even if their fate is in limbo.

The delay has outraged supporters of Mr. Trump’s who took his vows to rescind President Barack Obama’s program, called Deferred Action for Childhood Arrivals, very seriously.

“He’s really starting to anger…