WASHINGTON — Late-stage negotiations in Washington on a new $450 billion coronavirus aid package dragged past Monday’s hoped-for deadline, but the Trump administration and key lawmakers insisted a final pact is within reach.

President Donald Trump said he expects a Senate vote “hopefully” on Tuesday.

As talks continued, the contours of the deal appear largely set. Most of the funding, some $300 billion, would go to boost a small-business payroll loan program that’s out of money. Additional help would be given to hospitals, and billions more would be spent to boost testing for the virus, a key step in building the confidence required to reopen state economies.

The emerging draft measure — originally designed by Republicans as a $250 billion stopgap to replenish the payroll subsidies for smaller businesses — has grown into the second largest of the four coronavirus response bills so far. Democratic demands have caused the measure to balloon, though they likely will be denied the money they want to help struggling state and local governments.

The Senate met for a brief pro forma session Monday afternoon that could have provided a window to act on the upcoming measure under fast-track procedures requiring unanimous consent to advance legislation, but it wasn’t ready in time.

Majority Leader Mitch McConnell, R-Ky., set up another Senate session for Tuesday in the hope that an agreement will be finished by then.

The House has announced it could meet as soon as Wednesday for a vote on the pending package, according to a schedule update from Majority Leader Steny Hoyer, D-Md. The chamber is likely to have to call lawmakers back to Washington for a vote, which will present logistical challenges.

With small-business owners reeling during a coronavirus outbreak that has shuttered much economic activity, Treasury Secretary Steven Mnuchin — the administration’s point man in the talks with Democrats — said he was hopeful of a deal that could pass Congress quickly and get the Small Business Administration program back up by midweek.