WASHINGTON—President Trump in Tuesday’s State of the Union is expected to issue a plea for national unity at a moment of deep partisan divisions, with the Democratic-controlled House and the White House at an impasse over spending priorities while Mr. Trump threatens to do an end-run around both chambers of Congress to pay for a wall on the southern U.S. border.

Mr. Trump’s call for bipartisanship will be a contrast with his recent assertions that bipartisan efforts to reach a deal on border security are a “waste of time.”...