Senate Majority Leader Harry ReidHarry Mason ReidNo, it is not racist to question birthright citizenship McConnell rejects Democrats' 'radical movement' to abolish filibuster Harry Reid: 'Decriminalizing border crossings is not something that should be at the top of the list' MORE (D-Nev.) filed cloture on a House bill that the Senate will use as a vehicle to pass an unemployment insurance extension likely by Friday.

Reid filed the motion to end debate on H.R. 3979, the Protecting Volunteer Firefighters and Emergency Responders Act, which the House passed earlier this month.

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H.R. 3979 aims to exempt volunteer firefighters and EMTs from being considered full-time employees under ObamaCare mandates. Reid will use that bill as a way to send a five-month unemployment insurance (UI) extension bill back to the House.

Since Reid filed cloture, that would set up a vote as soon as Friday but Democrats are hoping to reach an agreement to hold the vote Thursday evening.

Sens. Jack ReedJohn (Jack) Francis ReedSenate Democrats push for arms control language in defense policy bill What the gun safety debate says about Washington Senators ask for committee vote on 'red flag' bills after shootings MORE (D-R.I.) and Dean HellerDean Arthur HellerThis week: Barr back in hot seat over Mueller report Trump suggests Heller lost reelection bid because he was 'hostile' during 2016 presidential campaign Trump picks ex-oil lobbyist David Bernhardt for Interior secretary MORE (R-Nev.) have put together a plan that would provide retroactive benefits to more than 2 million people who lost federal help after the program expired on Dec. 28.

The Senate has failed to pass two other UI extensions, but this time the legislation has five Republican cosponsors, meaning it could overcome the 60-vote threshold of a filibuster.

It would use several offsets to pay for the $10 billion cost of extending the benefits, including pension smoothing provisions from the 2012 highway bill, which were set to phase out this year, and extending customs user fees through 2024.

The Senate deal also includes an additional offset allowing single-employer pension plans to prepay their flat rate premiums to the Pension Benefit Guaranty Corporation (PBGC).

The measure would also prevent millionaires and billionaires from receiving the federal benefits.

The proposal also includes language pushed by Collins to strengthen reemployment and eligibility assessment (REA) and re-employment services (RES) programs, which provide help to unemployed workers when they enter their 27th week of benefits.

Despite the likelihood of Senate passage, Speaker John BoehnerJohn Andrew BoehnerScaramucci compares Trump to Jonestown cult leader: 'It's like a hostage crisis inside the White House' Israel should resist Trump's efforts to politicize support Lobbyists race to cash in on cannabis boom MORE (R-Ohio) has said he won’t consider the Senate deal because it doesn’t include job-creating measures. But Senate passage will put pressure on BoehnerJohn Andrew BoehnerScaramucci compares Trump to Jonestown cult leader: 'It's like a hostage crisis inside the White House' Israel should resist Trump's efforts to politicize support Lobbyists race to cash in on cannabis boom MORE to do something.

The emergency federal program kicks in once workers who continue looking for a new job have exhausted benefits, usually after 26 weeks.

— Vicki Needham contributed to this article.