WASHINGTON — The White House and Democrats rushed on Wednesday to reach agreement on emergency legislation to provide a first tranche of economic assistance to help Americans cope with the fast-moving coronavirus pandemic, with the hope of sending it to President Trump for his signature by the end of the week.

As the White House and Democrats were divided over what a broader economic stimulus package should look like, the two parties were trying to coalesce around the idea of a narrower short-term bill, while deferring discussion over other economic measures until after Congress returns from a weeklong recess. The bill would focus on paid leave, enhanced unemployment insurance, food assistance and help for small businesses.

By nighttime, though, there was still no deal, and House Democrats were pushing ahead with their own emergency relief package they were calling the “Families First Coronavirus Response Act.” They expect to bring it to the floor for a vote on Thursday before they leave Washington for a scheduled break.

The bill will include free coronavirus testing, paid emergency leave for workers, food security assistance, help to states overburdened by Medicaid costs and strengthened unemployment insurance, according to a joint statement issued Wednesday night by Speaker Nancy Pelosi and Senator Chuck Schumer, the minority leader.