President-elect Donald J. Trump on Wednesday appeared to soften his stance on whether to deport the more than 700,000 young people who entered the country illegally as children and were permitted to stay by President Obama.

“We’re going to work something out that’s going to make people happy and proud,” Mr. Trump told Time magazine. “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.”

The immigrants, who call themselves Dreamers, are likely to present Mr. Trump with one of the first major policy tests of his administration, as his campaign promise to take a tough stance on immigration clashes with lawmakers from both parties on Capitol Hill who have implored Mr. Trump not to deny these young adults protective status.

As Mr. Trump has tempered some of his more contentious campaign vows, including saying he would not seek to jail Hillary Clinton, his promise to take a tougher stance toward unauthorized immigrants is one his base of supporters is likely to demand that he keep. “He was not particularly interested in focusing on prosecuting Hillary Clinton any further because he was focused on health care and immigration,” Kellyanne Conway, a senior adviser to Mr. Trump, told Fox News on Sunday.

In the Time interview, part of the magazine’s naming of Mr. Trump its person of the year, the president-elect did not go into specifics or say whether he would reverse his promise to overturn Mr. Obama’s executive actions, including the 2012 Deferred Action for Childhood Arrivals, or DACA, which has shielded immigrants brought to the country illegally as children.

Undoing Mr. Obama’s protections for Dreamers would be one of the simplest moves Mr. Trump could take on immigration. He could withdraw the president’s executive order and let Congress address the young immigrants legislatively, which several lawmakers crave to do next year. Or Mr. Trump could do nothing, leaving the order in place.

At a meeting between the president-elect and Mayor Rahm Emanuel of Chicago on Wednesday, the topic of what to do with the young immigrants dominated. Mr. Emanuel, a former chief of staff to Mr. Obama, told reporters afterward that he discussed White House operations and immigration with Mr. Trump.