The left opposes the new law, arguing that it is unnecessary and aimed at reducing turnout among Democratic voters.

Senate Majority Leader Charles E. Schumer is maneuvering to pass a third bill this year using budget reconciliation, an obscure procedure that sidesteps the filibuster and lets Democrats ram through President Biden’s agenda without the support of a single Republican.

Mr. Schumer, New York Democrat, is proposing to expand the budget reconciliation process, which allows Congress to pass spending, revenue and debt measures with a simple majority of 51 votes.

Reconciliation measures are not subject to the filibuster, a Senate rule that requires most legislation to clear a 60-vote hurdle...