Congressional Democrats are taking a rare gamble on government shutdown politics they have decried in the past, wagering that health care is a winning cause with Americans.

Congress’ top two Democratic leaders left a White House meeting with President Trump, Vice President J.D. Vance and Republican leaders Monday pledging to maintain their blockade of Republicans’ stopgap funding bill to keep the government open through Nov. 21.

“This is a health care fight,” said House Minority Leader Hakeem Jeffries, New York Democrat. “But it’s also a fight to lower the high cost of living here in the United States of America.”

The Senate will vote on the House-passed Republican short-term spending bill for the second time Tuesday, just hours before the government shuts down at midnight. Republicans are urging enough Senate Democrats to join them in keeping the government open but know that outcome is unlikely.

“I think we’re headed into a shutdown because the Democrats won’t do the right thing,” Mr. Vance said. “I hope they change their mind, but we’re going to see.”

Democrats’ main request is permanently extending the COVID-19 expansion of Obamacare subsidies that cap out-of-pocket premium costs for buying health insurance on the public exchanges at 8.5% of household income.

The enhanced subsidies expire at the end of the year. Still, notices of increased premiums start going out Wednesday, and insurers are unlikely to adjust rates after open enrollment begins Nov. 1.

“He was not aware that the real effect of that starts Oct. 1, not Dec. 31,” Senate Minority Leader Charles E. Schumer, New York Democrat, said of Mr. Trump. “So it seemed from his body language and some of the things he said that he was not aware of the ramifications.”

Senate Majority Leader John Thune, South Dakota Republican, described the Democrats’ tactics as “purely and simply hostage taking.” He said Republicans are willing to negotiate with Democrats on extending the Obamacare subsidies “with reforms” but not under the threat of a government shutdown.

Democrats do not believe those negotiations will actually materialize.

“Later means never. They don’t want to do this, the Republican leaders,” Mr. Schumer said. “It’ll divide their party. Their right wing hates ACA altogether. But we have to get it done.”

“The way to do this is the president, who was really listening to us, tells the Republicans to do it,” he said.

Mr. Vance said he and Mr. Trump found some of the Democrats’ ideas “reasonable” but said Republicans would not agree to anything until the government is funded.

“We want to work across the aisle to make sure that people have access to good health care,” Mr. Vance said. “We are not going to let Democrats shut down the government, take a hostage, unless we give them everything they want.”

Democrats’ gamble may not pay off. No party that has forced a government shutdown to secure leverage over unrelated policy demands has been able to reopen the government with significant concessions in law.

More often, Republicans have driven the shutdown showdown. In 2013, Sen. Ted Cruz of Texas led a band of Republican rebels in shutting down the government to try to defund Obamacare, but they failed to achieve that goal.

Mr. Trump prompted the longest-ever government shutdown, a 35-day stretch that began in December 2018 and lasted into January 2019, over his demand for border wall funding. The government reopened without him securing any, although a later bipartisan agreement did provide a fraction of the wall funding Mr. Trump sought.

Democrats last delved into shutdown politics in early 2018 as they used the government funding deadline to try to force Republicans to agree to provide legal status for undocumented immigrants who came to the country as children under the Deferred Action for Childhood Arrivals program. The government was shut down for three days before Democrats agreed to reopen it with just the promise of a future vote on DACA, which ultimately failed.

Mr. Jeffries said Democrats do not trust Republicans not to undermine a verbal agreement and are seeking a more “ironclad” deal.

“Any bipartisan agreement by necessity has to have something in the legislation that makes clear to the American people that what we agree upon actually takes place,” he said.

In addition to an extension of the pandemic-enhanced Obamacare subsidies, Democrats are demanding that any government funding bill include provisions to stop the Trump administration from using rescissions and impoundment to ignore congressional spending directives.

“How could we negotiate a bipartisan agreement and then have the president unilaterally … undo it all?” Mr. Schumer said.

Republicans have insisted that their seven-week stopgap spending bill is a “clean” extension, meaning free of partisan riders, and is not open for negotiation. House Republican leaders canceled votes this week because their chamber has already passed the stopgap.

“If the Democrats make the decision to shut the government down, the consequences are on them,” Mr. Johnson said. “And I think it’s absolutely tragic.”

Part of the Democrats’ gamble is that they believe voters will blame Mr. Trump for the shutdown because Republicans control the White House and Congress.

Mr. Schumer said he brought that up with Mr. Trump during the meeting.

“I said, ‘You know the president gets the blame for this stuff.’ He admitted that,” he said.

White House Office of Management and Budget Director Russell Vought sent guidance to government agencies last week on a potential shutdown, suggesting department heads use it to fire government workers deemed nonessential to maintaining operations.

Republicans declined to comment about the potential firings after Monday’s meeting, but Mr. Vance vowed they will “keep essential services functioning.”

Mr. Vought said he wants the government to remain open but is preparing for a potential shutdown.

“We will manage it appropriately,” he said. “But it is something that can all be avoided.”