Mr. Kushner conceded in a briefing after the president’s speech that he did not see the proposal as a solution for the DACA program, which Mr. Trump moved to rescind in 2017.

“At this moment in time,” Mr. Kushner said, “this is a good path forward.”

Many conservatives did not share that view.

“Trump proposes amnesty,” the conservative commentator Ann Coulter said on Twitter. “We voted for Trump and got Jeb!” she added, referring to Jeb Bush, who challenged Mr. Trump for the Republican nomination in 2016 and supported a broad immigration overhaul that would have given undocumented immigrants a path to legal status.

Still, in the eyes of many White House officials, the prospect that Mr. Trump could use the proposal to shift blame for the shutdown and pressure Democrats to end the impasse was worth trying. Mr. Pence argued on Saturday that the speech was a “sincere effort” by Mr. Trump to break the logjam, and he and other White House officials suggested that the measure could attract enough support to succeed from centrist Democrats fed up with the shutdown and willing to side with Republicans.

But such a coalition did not appear to be forming, and courting one bears considerable risk for a president who is most comfortable when he is defying convention, eschewing compromise and being hailed as a hero by supporters who often equate bipartisan deal making with weak-kneed capitulation.

The vast majority of Democrats knocked the approach. While many of them have pressed for measures to protect DACA recipients and immigrants living in the United States under Temporary Protected Status enacted when their countries were destabilized by war or catastrophe, most regard the proposal he put forth on Saturday as woefully inadequate. It offers only three years of protections for the DACA recipients and those who hold T.P.S., which the Trump administration has also moved to end for several countries.

“This is not an amnesty bill,” Mr. Pence said. “There is no pathway to citizenship in this proposal.”