“It’s not clear what delaying this for six months means,” said Mark Krikorian, an immigration hard-liner who runs the Center for Immigration Studies who has supported the president’s actions to curtail immigration.
“He’s being pulled in a bunch of different directions, and because he doesn’t have any strong ideological anchor, or deep knowledge of the issue, he ends up sort of not knowing what to do,” Mr. Krikorian said. “I think the fact that they did nothing to it suggests that they had no idea what to do,” he added.
Mr. Trump’s frenzied weekend search for an alternative to abruptly ending the program was a fitting finale to his anguished deliberations over DACA since he took office. Aides have portrayed it as a difficult emotional decision for the president.
The main pressure to end the program is coming from Mr. Trump’s hard-line policy adviser, Stephen Miller; Mr. Session; and his still-influential former chief strategist, Stephen K. Bannon, who believe that nothing short of a complete and immediate shutdown of the program will fulfill the president’s campaign pledges.
They have warned the president that immediate action is required to head off the lawsuits against the program brought by a group of red state attorneys general.
Mr. Kelly, who is broadly supportive of the program, has been searching for a way to stave off the threat of the lawsuits. He was also motivated to create a compromise after hearing reports of young DACA immigrants assisting in disaster recovery efforts during two trips to Texas last week, according to an administration official.
The no-decision decision, some in the administration conceded, is reminiscent of Mr. Obama’s deferred pledge to close the detention center at Guantánamo Bay, Cuba — another campaign promise that proved difficult to fulfill but impossible to abandon.
DACA, by contrast, affects many more people. Any decision by Mr. Trump to wind down the program would immediately start the clock on throwing previously undocumented immigrants who obtained temporary reprieves and authorizations to work — known as Dreamers — into a vulnerable status that would make them subject to potential deportation.
“The federal government has the cellphone and home address of every DACA recipient,” said Todd Schulte, president of FWD.us, a progressive immigration reform group that has urged the president to retain the program.
“They grew up here, they work at nearly every major company in America, serve in the military and many are working on recovery efforts in Texas,” he added. “If DACA is repealed and no permanent legislation passed, they will all be fired and our government will begin the large-scale deportation of people raised in the United States, using information they volunteered to the government with the promise it would never be used against them or their families.”
On Sunday night, a group from several states with DACA recipients held a candlelight vigil outside the home of Mr. Trump’s daughter Ivanka and son-in-law, Jared Kushner.
A spokesman for Mr. Obama said he was likely to make a rare public statement opposing Mr. Trump if DACA is changed, though he will probably confine his comments to social media.
The delay would also throw congressional Republicans into limbo.
Mr. Trump’s staff had always intended to refer the issue to Congress, and Mr. Kelly began discussing the possibility when he was in the cabinet, two senior White House official said. But they never planned to toss what one of the officials described as an unpinned hand grenade at Capitol Hill Republicans quite this soon.
Doing that, however, fits a pattern that Mr. Trump has established in recent months of postponing consequential decisions on contentious topics, like health care reform, leaving it to senior members of his administration or Republican congressional leaders to articulate a final position while seeking to deflect blame for the results.
Hill Republicans, especially House Speaker Paul D. Ryan, had urged the president not to introduce immigration reform — an issue that has splintered the party in recent years — until lawmakers had passed an increase in the debt ceiling, and spending and tax reform bills.
Moderates, including Mr. Kushner, and the National Economic Council chairman, Gary D. Cohn, had urged the president to reach beyond his hard-right populist base to embrace a program that enjoys significant public support, even among Republicans. Business leaders, among them political allies like the media mogul Rupert Murdoch and Stephen A. Schwarzman, chairman and chief executive of the Blackstone Group, believe any move to limit legal immigration limits the work force and hurts the country’s international reputation.
But the moderates in Mr. Trump’s midst, Mr. Cohn in particular, are somewhat less influential these days, after several expressed their disgust at the president’s response to the racial riots in Charlottesville, Va., last month.
Mr. Sessions, who has experienced his own deep freeze with Mr. Trump over the Russia investigation, has regained some of his influence, pressing Mr. Trump to end the DACA program before the courts force his hand. Mr. Miller, a former Sessions aide, has cast Mr. Trump’s actions as a key test of the president’s commitment to the economic nationalist agenda he ran on in 2016.
Mr. Trump initially embraced their call for urgency.
Now that sense of urgency is being felt by supporters of DACA in both parties — and they may not be willing to wait six months to reassure endangered immigrants that they will have a place in their adopted country.
Representative Mike Coffman, a Colorado Republican who represents a heavily Latino district and faces a tough re-election bid, said Monday that he planned to push a legislative maneuver to get a vote on an extension of DACA co-sponsored by Representative Luis V. Gutiérrez, an Illinois Democrat.
“There are a number of Democrats who are co-sponsors on it, so I think it would be tough to play political games with the bill,” Mr. Coffman said on Monday. “I will get heat from a lot of conservatives in my district.”