Activists appeal for a $15 minimum wage near the Capitol in Washington, Thursday. The $1.9 trillion COVID-19 relief bill, which has passed in the House, is now being debated in the Senate.

WASHINGTON — Democrats’ efforts to include a minimum wage increase in their $1.9 trillion COVID-19 relief bill seemed all but dead Monday as Senate leaders prepared to begin debate on their own version of the House-passed aid package.

Top Democrats have abandoned a potential amendment threatening tax increases on big companies that don’t boost workers’ pay to certain levels, Senate aides said. Four days after the chamber’s parliamentarian said Senate rules forbid inclusion of a straight-out minimum wage increase in the relief measure, Democrats seemed to have exhausted their most realistic options for quickly salvaging the pay hike.

“At this moment, we may not have a path, but I hope we can find one” for pushing the federal pay floor to $15 an hour, said No. 2 Senate Democratic leader Richard Durbin of Illinois.

Senate Democrats hope to unveil their version of the broad relief package and begin debate as early as Wednesday. Congressional leaders want to send President Joe Biden the legislation combating the pandemic and bolstering the economy by March 14, the date emergency jobless benefits that lawmakers approved in December expire.

The bill is Biden’s biggest early legislative priority. It looms as an initial test of his ability to unite Democrats in the Senate — where the party has no votes to spare — and risks damage to his influence should he fail. Republicans are strongly against the legislation and could well oppose it unanimously, as House GOP lawmakers did when that chamber approved the bill early Saturday.

The Senate is divided 50-50 between the parties, with Vice President Kamala Harris able to cast only tie-breaking votes.