Senate Democrats are jettisoning a proposal to use the tax code to penalize corporations that don’t raise the minimum wage for their lowest-paid workers in an effort to keep President Joe Biden’s broader stimulus plan on track for quick passage, according to two people familiar with the matter.

Democrats were left seeking alternatives after the Senate parliamentarian ruled Thursday that raising the federal minimum wage to $15 an hour as part of the $1.9 trillion pandemic relief legislation failed to qualify under fast-track budget rules that Democrats are using to pass the stimulus without Republican support.

The Senate could begin consideration of the stimulus measure as soon as Wednesday — with final votes as soon as late Thursday — pending full Democratic support and sign off from the parliamentarian.

Senate Finance Committee Chairman Ron Wyden and Budget Committee Chairman Bernie Sanders proposed tax penalties on big companies that pay low wages along with incentives for smaller companies as an alternative. But it became clear over the weekend that getting all 50 Senate Democrats to agree on specific language would risk missing the March 14 deadline for extending expiring supplemental unemployment benefits, said one of the people, who spoke on condition of anonymity to describe private discussions.

The decision removes a major complication in the Senate and could speed approval of the rest of the package in the chamber. In addition to getting the backing of all 50 senators who caucus with Democrats, the tax language also would have had to pass muster with the parliamentarian, the House and the administration and be signed by Biden -- all in a two-week period.

Still, dropping the minimum wage increase from the stimulus could create other headaches for both Senate Majority Leader Chuck Schumer and House Speaker Nancy Pelosi.

In the House, the stimulus legislation with the $15 an hour minimum wage narrowly passed on Saturday by 219 to 212, without any Republican support. Pelosi lost two moderate Democrats on that vote, and she can’t afford many more defections.

Pelosi said Friday she’s “absolutely” confident she could pass a stimulus bill without a minimum-wage provision in it, but that remains to be seen.

Progressives in the chamber have made the minimum wage hike a signature issue as they angle to set the party’s future direction. Although there hasn’t been an explicit threat to scuttle the broader stimulus bill, which includes the extra unemployment benefits and $1,400 direct payments to many Americans, the conflict is bound to increase tension between the two wings of the party with Biden in the middle.

Progressive lawmakers on Monday sent Biden and Vice President Kamala Harris a letter organized by Representative Ro Khanna demanding that they set aside the parliamentarian’s ruling and include a minimum-wage increase in the Covid-19 relief package.

New York Representative Alexandria Ocasio-Cortez, the leader of a group of young, vocal progressives, signed the letter and is among those urging Senate Democrats to get rid of a rule that gives the minority power to block most legislation.

“Really our options right now, at least our immediate options on this specific issue, is to do something about this parliamentary obstacle or abolish the filibuster,” she said on Friday.

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