WASHINGTON—President Donald Trump’s call for lawmakers to embrace what he termed a compromise proposal on immigration got a chilly reaction Wednesday from both Democrats and Republicans, who are struggling to make progress in their negotiations.

In his State of the Union speech Tuesday night, Mr. Trump highlighted the details of his plan, which include funding for a wall on the southern border, offering a path to citizenship for 1.8 million people brought to the country illegally when they were young, limiting a family-based...