Speaker Nancy Pelosi has ruled out paying for a border wall, even at a much lower funding level, despite suggestions by Mr. Trump that the government shutdown could last “months or even years.”

Administration officials have been scrambling to tackle the legal and logistical problems involved with steering previously allocated funding toward any project that can be described as a wall, a fence or a barrier.

The definition of a wall has grown ever more elastic.

The White House and homeland security officials have been pushing the Pentagon to continue to use military troops at the border — mainly to install, extend and repair a section of concertina wire used to stop immigrants from entering unmanned sections of the southwest border.

Mr. Trump tweeted last month that “the Military will build the remaining sections of the Wall” — a reference to several hundred miles of fencing along the border.

The temporary troop deployments are just one of many recommendations that the homeland security secretary, Kirstjen Nielsen, and her staff have made to the Pentagon.

One possible solution, in the absence of new congressional allocations, is figuring out a way to steer contracts already planned for Army Corps of Engineers projects toward some kind of barriers. But lawyers at the Homeland Security and Justice Departments have yet to determine if doing so could withstand court challenges, two senior administration officials said.

Mr. Trump spent the holidays at the White House to demonstrate his determination to secure new border funding. Yet he has struggled to dramatize his cause, as stories of federal employees and other Americans suffering from the effects of the shutdown pile up.