Mr. McConnell, 76, is the longest-serving Senate Republican leader in history and is a wily tactician. While he is not generally considered vulnerable, his popularity is lagging at home. A poll by Western Kentucky University last spring found that only 30 percent of Kentuckians approve of his job performance. Sticking with Mr. Trump, whose approval ratings top 50 percent in Kentucky, is in his own political interest.

But if he wants to hang onto his job as majority leader, Mr. McConnell must also be mindful of the political fortunes of Republicans seeking re-election in states won by Mrs. Clinton in 2016 or by Democrats in November. A prolonged government shutdown is the last thing those lawmakers need. And even some Republicans up for re-election in states won by Mr. Trump, like Senator Shelley Moore Capito of West Virginia, sound uneasy.

“A shutdown, in my view, is a no-win proposition,” Ms. Capito said, adding that she heard complaints from Transportation Security Administration workers as she was returning to Washington for the start of the new Congress this week. Ms. Capito is the chairwoman of the appropriations subcommittee with jurisdiction over the Department of Homeland Security; the panel has already passed a bill funding the department, including $1.6 billion for border security including fencing — but with no money for Mr. Trump’s wall.

“It is just a lot of unneeded stress on a lot of people,” she said of the shutdown.

“I would like to see it resolved soon,” said Senator Joni Ernst, Republican of Iowa, who also faces re-election in 2020. She repeated herself for emphasis: “I would like to see it resolved soon.”

For Mr. McConnell, the shutdown fight presents a dynamic that is likely to become familiar over the next two years, as House Democrats rush to pass long-sought liberal policies and, in many cases, try to use the chamber to highlight Republicans’ opposition to legislative changes they believe are overwhelmingly popular. Mr. McConnell has already been burned once, having negotiated and passed through the Senate a plan to avoid the shutdown in the first place only to have Mr. Trump pull his support at the last minute.

“He faces that reality now on every issue: What’s the White House going to do with this?” said Senator Richard J. Durbin, Democrat of Illinois and Mr. Schumer’s No. 2. He predicted that Mr. McConnell’s approach — to simply stand in the way so that Mr. Trump is not forced to use his veto pen — would only change if Republican senators up for re-election begin to fear political costs of carrying the president’s water.



Senator John Cornyn of Texas, Mr. McConnell’s former No. 2, compared Mr. McConnell’s task to “threading a needle.” Mr. McConnell’s former chief of staff, Josh Holmes, said it was considerably more simple than that.