WASHINGTON >> Senate Republicans and the White House on Monday threw their support behind a substantial cut in jobless aid for tens of millions of Americans laid off amid the pandemic, proposing a weekly reduction of $400 to a benefit that has cushioned the nation’s economy even as coronavirus cases continue to rise across the country.

The proposal was part of a $1 trillion opening bid that would have to be reconciled with Democrats, who are pushing a recovery package that would spend three times as much and extend the $600 per week in extra unemployment aid through the end of the year.

Economists say the money, slated to expire this week, has provided a crucial economic buffer for the unemployed, and that lowering the payments could have a cascade of damaging effects across the economy. But Republicans contend that it is too generous, discouraging Americans from returning to work and hampering a recovery.

The Senate Republicans’ decision to embrace the decrease reflects the predicament in which they find themselves amid a worsening pandemic and continued economic recession, little more than three months before Election Day. With a small but vital bloc of conservative senators opposed to providing any more federal coronavirus aid, the party has struggled to agree on how to stabilize the battered economy, leaving Democrats with crucial leverage for an intense set of negotiations over the relief package.

Even as Republicans rolled out their proposal Monday evening, Mark Meadows, the White House chief of staff, and Steven Mnuchin, the Treasury secretary, were huddled in House Speaker Nancy Pelosi’s Capitol office suite, meeting with top Democratic leaders in a reflection of their influence in the talks.

With the two sides far apart, it appeared unlikely that they could bridge their differences in time to avert the lapse Friday of the supplemental jobless aid, nor was it guaranteed they would be able to do so at all. That left uncertain the fate of President Donald Trump’s best hope of injecting one last shot of stimulus into the economy before the general election in November.

Complicating the picture, Republicans and the White House continued to bicker over the contents of the package even after it was announced, with Sen. Mitch McConnell, R-Ky., the majority leader, appearing surprised that it included funding for a new FBI building that has long been an obsession of Trump’s.

“I don’t think there is funding, is there?” McConnell said to reporters who asked about the money, which is designated as a coronavirus-related emergency in the draft bill. Assured that it was, he said the administration “will have to answer the question on why they insisted on that provision.”

Republicans had hoped to avoid this situation altogether, knowing that many in their ranks had grown exhausted with the torrent of federal spending — nearly $3 trillion — that Congress approved in rapid succession in early spring. They resisted passing another package, gambling that if they waited, the virus would dissipate and the economy would rebound, and that they could push through a bare-bones package.

Instead, they are now staring down the beginning of the school year with skyrocketing cases and record unemployment levels, with many in their ranks unwilling to pour any more money into the economy. Their proposal spends more than many Republicans are likely to support, and it will most likely grow as Democrats place their stamp on it.

The package of bills rolled out Monday included a new round of $1,200 direct payments to Americans earning $75,000 or less per year. In line with Trump’s demands, it would reserve tens of billions of dollars in federal funding for schools that reopen for in-person instruction.

It would limit legal liability for businesses that open amid the pandemic, a top priority of business groups in Washington, for coronavirus-related episodes that take place through October 2024. It establishes a tax credit for companies to reconfigure their workplaces to promote safety from the virus, and it would expand tax credits for employers that hire and retain workers amid the outbreak. The proposal would also offer tax certainty to Americans who work in one state and live in another, and are facing the prospect of paying income taxes in multiple states if they were forced to work from home.

The package would both extend government aid for small businesses through the Paycheck Protection Program, which was established in March, and narrow the set of companies eligible to receive it. It would also create an alternate source of aid for businesses in low-income, high-poverty areas: a 20-year loan, with an interest rate of 1%, that would give those businesses enough cash to replace up to two years of lost revenues.

It also includes a bipartisan proposal to force Congress to consider future deficit and debt-reduction measures.

Meadows and Mnuchin sought to leave an indelible mark on the package on behalf of Trump, spending a weekend on Capitol Hill meeting with Senate staff — an unusual step for senior Cabinet officials — to hammer out the technical details of the unemployment proposal.

While the two men ultimately agreed to drop demands for a payroll tax cut — a presidential priority dismissed by members of both parties — they succeeded in securing $1.75 billion for the design and construction of a new building for the FBI Headquarters across from Trump’s luxury hotel in downtown Washington, in which he has repeatedly shown a personal interest.

The proposal to cut the jobless aid by two-thirds is likely to be among the most bitterly contested issues in the negotiations to come.