Mr Trump and Republican leaders in Congress have said they are working on a plan that will address the status of the roughly 750,000 immigrants currently protected by the Deferred Action for Childhood Arrivals program, or DACA. The program allows young people who were brought into the country illegally as children to stay and obtain work permits.
Neither the president nor GOP leaders have disclosed details on their discussions, although both have suggested those currently protected under the program won't face immediate deportation. Whether they will be allowed to continue to work remains unclear. Mr Trump said this past week he intends to reveal a proposal within a month.
"They shouldn't be very worried," Mr Trump told ABC News. "I do have a big heart. We're going to take care of everybody. ... Where you have great people that are here that have done a good job, they should be far less worried."
Mr Trump's delay and the tone of his remarks were a striking shift from the campaign, when he promised to quickly end the program and labeled it amnesty.
"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," Mr Trump said in August.
His current approach appears to be a concession to Republican leaders in Congress who have called for a less aggressive stand on an issue that has pushed some Latino voters away from the party.
House Speaker Paul Ryan has argued for a more compassionate approach in dealing with these young immigrants. He has discussed the issue privately with the president and recently said congressional Republicans had been working with his team on a solution.
DACA was arguably the most significant and high-profile change Mr Obama made to immigration policy, one he said he made only after Congress failed to enact a broader immigration overhaul. The status of the immigrants, some of whom have little or no connection to the country where they were born, has been a focus of advocates' anxiety and activism since Mr Trump's election.The program would be relatively simple for Trump to reverse. The policy does not require an executive order.
As of Friday afternoon, the US Citizenship and Immigration Services said it was still processing DACA-related work permits.
Mr Trump's White House is thought to be divided on the how to handle the issue.
Those who have pushed Mr Trump to embrace more restrictive immigration policies, including policy guru Stephen Miller, are said to prefer a harder line. On the other side are those in the White House considered more moderate, including chief of staff Reince Priebus, who said recently that the White House is planning to work with House and Senate leadership "to get a long-term solution on that issue."
Mr Trump faces pressure from GOP hard-liners, including Iowa Representative Steve King, who has chafed at signals from Mr Trump aides that immigrants covered by DACA are not a priority for deportation.
Mr King said Mr Trump risks a backlash from his political base if he doesn't act swiftly "because it was a clear and definitive promise that he made."
"And when you hear these kinds of statements coming out of the chief of staff and some of these statements that echo pretty closely out of the speaker of the House, it gives real pause to rule of law conservatives," he said.
Advocates on both sides are expecting a compromise in which the president perhaps ends DACA and then works with Congress on a permanent solution that allows these immigrants to stay in the country.
Mark Gonzales, the president of the Hispanic Action Network, said Mr Trump's actions this past week have "definitely shaken up our Hispanic community and the immigrant community in particular." But he said that he's confident from Mr Trump and his aides' rhetoric that young immigrants currently protected by DACA don't have to be alarmed.
"He's making clear that even though they're going to repeal DAPA and DACA, the Dreamers are not the ones they're going to be coming after," said Mr Gonzales. "They're going to find a way to deal with the Dreamers in the way that is not deportation. That's good news to our ears."
Mark Krikorian, the executive director of the conservative Center for Immigration Studies, which advocates for tighter restrictions on immigration, said he's open to a compromise in which, for instance, those previously covered by DACA receive green cards in exchange for other concessions.
"Suspending DACA processing is an extremely simple, clear-cut thing. And if they haven't done it by now, there's some reason for it," said Mr Krikorian.
Young people covered by the program have been glued to their televisions anytime Trump takes action, said California lawyer Sergio Garcia, who estimates he's handled more than 15,000 DACA applications.
There's "a lot of anxiety, uncertainty of not knowing what tomorrow holds," he said.