Senate Majority Leader Mitch McConnell Addison (Mitch) Mitchell McConnellFEC flags McConnell campaign over suspected accounting errors Poll: 59 percent think president elected in November should name next Supreme Court justice Mark Kelly: Arizona Senate race winner should be sworn in 'promptly' MORE (R-Ky.) is teeing up a first vote on a White House-backed criminal justice bill for early next week.

McConnell, wrapping up the Senate's work for the week, scheduled a procedural vote for 5:30 p.m. on Monday.

It will mark a crucial test of the bill's backing, where supporters will need to put up 60 votes to advance the bill. If they're successful, a final vote is expected to take place as late as Wednesday.

Supporters say they have at least 70 votes for the measure, though official cosponsors are at 35, according to Congress.gov

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The bill merges a House-passed prison reform measure with a handful of reforms to sentencing laws. Supporters rolled out a new version of the bill this week with changes aimed winning over more Republican support. The changes include adding additional crimes to a list of offenses that exclude an individual from the bill's earned time credits, which can shave time off a sentence.

Though the bill has broad support among both parties, it's run into vocal opposition from conservative senators who worry that it will allow repeat felons out of jail.