Last month, the Trump administration indicated that it would increase military spending by $54 billion while cutting nonmilitary spending by the same amount. The plan deeply slashes money for the State Department, foreign aid, the Internal Revenue Service and the National Endowment for the Arts.

A draft budget request that surfaced last week also showed that the Trump administration was considering making cuts to the Coast Guard, the Transportation Security Administration and the Federal Emergency Management Agency. Much of those funds would go toward paying for the wall.

Still, none of this will happen without the assistance of some Democrats.

Republicans, who have 52 seats in the Senate, will need 60 votes to pass a budget. Resistance to military spending increases from Republican fiscal hawks in the House could mean that some Democrats will be needed there, too. And Democrats have been steadfast in their insistence that they will not lift the caps on military spending imposed by the Budget Control Act sequester unless spending on domestic programs is allowed to increase by the same amount.

Democrats flirted with a government shutdown over health benefits to retired miners last year before backing down at the last minute.

Members of the party point out that their approach is different from the one taken by Republicans during the Obama years because they are merely seeking a clean budget bill, not using the threat of a shutdown to extract policy concessions from the party in power.

Mr. Schumer said that he would not rule out approving funding for a wall in 2018, but that Mr. Trump must first explain how he plans to use eminent domain to acquire land along the border, the effects of the construction on the Native Americans and how he plans to persuade Mexico to reimburse the United States for the costs.