“It doesn’t help us to pass the Dream Act and have it be a separate bill that never makes it to the president’s desk or gets vetoed on the president’s desk,’’ he said.

The arrival on Wednesday of Doug Jones, the newly elected Democrat of Alabama, leaves Republicans with an exceedingly narrow Senate majority, 51 to 49. The Senate’s major legislative accomplishment of last year, a far-reaching overhaul of the tax code, was muscled through using procedural maneuvers that allowed for a simple majority, party-line victory.

But Republicans cannot use such maneuvers on a spending bill, which will require 60 votes for passage. That gives Democrats an advantage, said Senator Chuck Schumer, Democrat of New York and the minority leader.

“Republicans know that a government shutdown falls on their back, because they’re in charge of House, Senate and the president, and for the first time they need Democrats to accomplish their goals,” Mr. Schumer said.

Wednesday’s meeting is supposed to focus on spending levels, which were fixed in the 2011 legislation that ended another fiscal showdown. It will include Ms. Pelosi and Mr. Schumer, Mr. McConnell and Speaker Paul D. Ryan, as well as Mr. Trump’s budget director, Mick Mulvaney, and his liaison to Congress, Marc Short.

Both Mr. Schumer and Ms. Pelosi said Democrats would use the session to insist on “parity,” meaning that military and domestic spending must be increased by the same amount. Republicans are far more focused on building up the military.

“This idea that we’ve got to increase domestic spending dollar for dollar, it has no basis in reality,” said Representative Tom Cole, Republican of Oklahoma. “It’s simply a way to try to use the military to blackmail the government to spend more money domestically.”