WASHINGTON—Congressional leaders struck a deal with the White House Tuesday to send hundreds of billion of dollars in fresh aid to small businesses and hospitals, the federal government’s latest effort to keep pace with the twin economic and public health crises created by the coronavirus pandemic.

The Senate on Tuesday evening passed the $484 billion bill by a voice vote, sending it to the House for an approval expected Thursday. President Trump said on Twitter he supported the legislation, whose final components were hammered out between top White House officials and congressional leaders in the early morning hours Tuesday.

The package, which lawmakers dubbed an interim emergency bill, also includes funding to ramp up the country’s testing for the new coronavirus, but doesn’t include funding sought by Democrats for hard-hit state and local budgets, which instead was pushed off to the next round of stimulus negotiations. Top Republicans signaled that concerns over the mounting debt would play a bigger role in talks about future stimulus aid, setting up a sharp divide with Democrats worried that Congress has done far from enough.

Congress has operated in emergency mode during the outbreak, last month passing a $2.2 trillion package by consensus with minimal debate. Amid growing pressure from rank-and-file members, House leaders agreed Tuesday to hold a vote later this week on allowing proxy voting, which would allow members to cast votes without meeting in person.

The agreement reached Tuesday caps a brief but bitter fight over funding for the Paycheck Protection Program, a highly sought-after small-business loan fund that both parties wanted to see replenished after it depleted its initial $350 billion allocation last week. Lawmakers sparred for nearly two weeks over what else should be funded in the relief package, with the GOP pushing to quickly pour more funds into the small-business program and Democrats holding out for money for hospitals, testing, food stamps and state and local governments.