The president’s attempts to blame Democrats for the shutdown has gained little traction with the public. About 47 percent of adults hold Mr. Trump responsible for the shutdown, according to a Reuters/Ipsos poll released on Thursday. In contrast, about 33 percent blame the Democrats in Congress. The poll was conducted Dec. 21 through 25, mostly after the shutdown went into effect.

Despite Mr. Trump’s assertions to the contrary, most Democrats support increased funding for border security. Many Senate Democrats, for instance, voted to allocate tens of billions of dollars for security enhancements in 2013, including roughly doubling the size of the border patrol, creating new electronic monitoring systems and even some additional physical barriers at certain points along the border.

But that money was in the context of a comprehensive immigration overhaul that would have tried to address systemic problems. By contrast, they view Mr. Trump’s border wall as a costly and ineffective proposal. What is more, they have pointed to government accounting documents that show the Trump administration has spent only a fraction of the money allocated by Congress last year for a physical barrier along the border.

With no endgame in mind, Mr. Trump appears to be boxed in between Democrats who will control the House and have key votes in the Senate and his supporters on the hard right, who have grown more vocal in criticizing him. After administration officials signaled this month that the president was willing to back off his demands for wall funding, the conservative personalities Ann Coulter and Rush Limbaugh accused the president of being “gutless” and surrendering to Democrats.

To avoid an immediate hit from his base, Mr. Trump appeared to embrace a short-term strategy, celebrating a premature victory when the House passed a short-term funding bill last week that included the money he wanted for more border security.

Typically in spending negotiations, lawmakers join the president in hammering the opposition party over their position. But other than Representative Mark Meadows of North Carolina, who is a close ally of the president, and a handful of arch-conservative House members, most Republican lawmakers have stayed out of the fray. Even Senator Mitch McConnell of Kentucky, the majority leader, has held off, saying last week that it was up to Democrats and Mr. Trump to reach a deal and only when they did would he hold a vote.

“It’s clear that we on the Republican side do not want to vote for a bill that the president won’t sign,” Senator Pat Roberts, Republican of Kansas, told reporters on Thursday after presiding over a minutes-long Senate session.