“Why is the president ignoring the sanctions?” asked an exasperated Senator Bill Nelson, Democrat of Florida, a veteran of three decades in Congress. “I’ve never seen it like this.”

Republicans, in praising the president for bold leadership, say Democrats bear some responsibility for finding bipartisan solutions and need to get past their animosity for Mr. Trump if they want a deal on immigration.

“If Democrats refuse to negotiate now after what President Trump put on the table, perhaps they should ‘migrate’ out of Congress,” said Representative Jeb Hensarling, Republican of Texas.

The ongoing hostility is not without consequences. Government funding runs out again on Feb. 8 and delicate negotiations are continuing in an effort to reach an immigration deal by early March at the latest. The parties need to have some level of trust and willingness to work together to resolve these difficult issues, since both spending and immigration measures will require votes from both sides of the aisle to become law.

To that end, Mr. Trump said in his speech that he was “extending an open hand to work with members of both parties.” He urged lawmakers “to seek out common ground, and to summon the unity we need to deliver for the people we were elected to serve.”

But to many Democrats, that appeal rang hollow given the past year that saw the Republican effort to unravel the Affordable Care Act and Mr. Trump’s tendency to back away from tentative compromise agreements with Democratic leaders after objections from Republicans.