At the White House, President Trump is posting A.I.-generated memes about the government shutdown, depicting his wonky budget director dressed as the Grim Reaper and ready to visit death on the federal bureaucracy.

In the Senate, Democrats show no sign of backing down from their demands in the shutdown fight, while Senator John Thune, Republican of South Dakota and the majority leader, has given verbal shrugs to reporters who ask about the status of his nonexistent negotiations with the other party about how to bring the crisis to an end.

“I don’t know that there’s a lot to sort out,” Mr. Thune said on MSNBC on Friday, before sending senators home for the weekend.

And in the House, Speaker Mike Johnson has canceled votes for this week, telling his members they could stay home for the third straight week given the shutdown logjam.

With Mr. Trump and lawmakers having made no progress on a deal that would reopen the government, one thing was clear as the shutdown headed into its second workweek: There was little sense of urgency in Washington about cleaning up a mess that has thousands of federal workers facing furloughs and possible layoffs, and could disrupt critical federal programs.

It all reflects the reality of two parties so convinced that they have the political advantage in their partisan battle that a shutdown has seemed inevitable for weeks, and a quick resolution feels out of reach.