WASHINGTON—The White House has thrust a new set of proposals into talks to avoid shutdown of the government next week, while also seeking to revive a health-care overhaul that had collapsed last month.

With less than a week to pass legislation funding the government for the rest of the fiscal year, negotiations are beginning to take shape. Democrats are demanding that the legislation include money for insurance companies, without which fragile insurance markets could implode, while the White House in return wants additional money for defense, the border wall and border enforcement.

Failure to extend the funding would trigger a partial government shutdown on April 29, the 100th day of Donald Trump’s presidency.

Republican leaders will need Democratic votes in the Senate, and likely in the House, to pass a spending bill, giving the minority party unusual leverage in negotiations. Discussions now hinge on Democratic demands that the government continue payments that help support Affordable Care Act insurance plans. The money, known as “cost-sharing” payments, helps insurers lower costs for low-income consumers.

On Thursday, White House budget director Mick Mulvaney said the administration was proposing, in return, increased defense spending, money to expand a border wall and funding to hire more immigration officers. Mr. Mulvaney said the White House was willing to give Democrats a commitment to continue payments to health insurers, provided that the Democrats agree to some White House conditions.