The Senate as early as Wednesday could vote on a "Sense of the Senate" bill that says taxpayers earning $1 million or more each year should "make a more meaningful contribution to the deficit-reduction effort."

The bill, titled the "Sense of the Senate on Shared Sacrifice," was introduced last week by Senate Majority Leader Harry Reid Harry Mason ReidGOP senators confident Trump pick to be confirmed by November Durbin: Democrats can 'slow' Supreme Court confirmation 'perhaps a matter of hours, maybe days at most' Supreme Court fight pushes Senate toward brink MORE (D-Nev.). Reid filed at cloture motion on the bill on Tuesday afternoon, meaning a vote to end debate could take place as early as late Wednesday or, more likely, Thursday.

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The bill has no specific recommendations on how much taxes should be raised on high-income earners, and is simply a recommendation that these taxpayers pay more. Because the 60 votes needed to end debate are unlikely to materialize, the vote will likely be used by Democrats as a way to show Republican resistance to new tax hikes.

Democrats might also try to use the vote as leverage in negotiations on how to raise the debt ceiling by showing that there is support for a tax increase. In those talks, which are expected to continue this week, Democrats have said taxes on the wealthy and on oil companies should be part of the equation for reducing the deficit. Republicans have so far rejected this, and argue that an agreement needs to focus solely on spending cuts.

Reid's bill, S. 1323, tries to make the case for higher taxes on the wealthy by arguing that the segment can afford it. Among other things, it notes that four-fifths of the income gains over the last quarter-century have accrued to the top 1 percent of earners.

The resolution does not appear to be on track to be offered as an amendment to the Libya resolution, and a GOP aide said that if it were offered, it would violate Reid's demands that amendments be germane.

To break from the Libya resolution to vote on the sense of the Senate would require unanimous consent, and so far it does not appear the GOP is yet ready to grant that in exchange for votes on a GOP alternative, sources said.

— This story was updated at 2:21 p.m. to add Reid's decision to file cloture on the bill.

Erik Wasson contributed.