WASHINGTON—As unemployment rises and the first wave of federal economic relief nears expiration, lawmakers are increasingly looking to expand an existing wage subsidy to keep workers on payrolls and help businesses stay afloat.

The $3 trillion package passed by the House this month features an expanded wage subsidy, known as the employee retention tax credit. That proposal, which would add about $194 billion to a $55 billion tax credit created in March, is gaining bipartisan support even as lawmakers clash over other legislation…

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