In a tweet Monday night, he added that “some things NEVER get better” over time: “You have Walls and you have Wheels. It was ALWAYS that way and it will ALWAYS be that way! Please explain to the Democrats that there can NEVER be a replacement for a good old fashioned WALL!”

And Representative Mark Meadows, Republican of North Carolina and the chairman of the House Freedom Caucus, blasted the Democrats’ proposal. “Nancy Pelosi’s newest funding proposal doesn’t represent any serious attempt to secure our border or find a compromise,” Mr. Meadows wrote on Twitter. “A $1.3 billion Democrat wish list that includes zero money for a border barrier is a non-starter and will not be a legitimate answer to this impasse.”

The shutdown standoff will give Americans their first glimpse at how Washington will work in a new era of divided government. It will also be the first leadership test for Ms. Pelosi, who anticipated a string of celebrations as she reclaims the gavel as the nation’s first female speaker of the House. She has continued to insist that Democrats will not cave on wall funding.

“He’s not going to get a wall,” Ms. Pelosi said in a recent interview. “But he has to recognize, we are the first branch, Article 1, the legislative branch, and we’re here not as a rubber stamp to the executive branch but are coequal to him.”

The shutdown is affecting parts of nine cabinet-level departments, and as it drags on, anxieties are rising among the 800,000 federal employees who are either furloughed or working without pay. In one ominous sign, the federal Office of Personnel Management has posted a sample letter for mortgage companies and creditors on its website, giving guidance to federal employees on how to negotiate reduced payments.

Democrats say their proposed measures would create a month of breathing space for the president to negotiate with them. But Mr. Trump would lose much of his leverage in the border fight if he accepts their terms.

The Democrats’ two-pronged plan will also complicate life for Mr. McConnell, who has said he will not bring up any measure that does not have the president’s support. Once the bills pass the House, as expected, Mr. McConnell will have to decide whether to put one, both or neither on the Senate floor for votes.