Democrats framed Tuesday’s vote as a referendum on protecting the separation of powers and Congress’s constitutional right to determine federal spending levels — an argument that appealed to several conservatives in both chambers.

Representative Joaquin Castro, Democrat of Texas and the author of the resolution, warned Monday at a news conference that without congressional interference, Mr. Trump would “try this again on other issues,” while Speaker Nancy Pelosi of California expressed confidence that her Republican colleagues’ belief in the separation of powers would push them to support the one-page resolution.

“This isn’t about the border,” she said. “This is about the Constitution of the United States.”

But Democrats were also making a less lofty case to wavering Republicans. They circulated a list of all of the possible military construction projects in each district that could lose money shifted instead to Mr. Trump’s wall.

Several lawsuits have already been filed to challenge the merits of the declaration, but the easiest way for Congress to counter it is through the resolution of disapproval, authorized by the National Emergencies Act of 1976. Once it passes the House, the Senate is required under the law to take it up within 18 days.

On Monday, Mr. Trump trained his attention on the Senate, where only four Republican votes are needed to pass the measure, should Democrats remain united, as expected. The president warned Republicans, via Twitter, not to “fall into the Democrats ‘trap’ of Open Borders and Crime!”

Several conservative senators have expressed concern that Mr. Trump’s declaration is setting a precedent that could be used by a Democratic president determined to secure funds that Congress will not give.

Others have balked at the prospect of siphoning money away from military projects. Senator Charles E. Grassley, Republican of Iowa — who said he was “leaning no” on voting for the resolution — suggested that Congress review the power to declare national emergencies granted to the president under the National Emergencies Act.