Rep. Bobby Scott, D-Va., who will likely lead the House Committee on Education and the Workforce next year, has backed legislation to gradually increase the minimum wage to $15 per hour. House Minority Leader Nancy Pelosi, the favorite to become speaker in January, supported the bill. Scott would move to raise the wage floor in one of his first acts as chairman, and has no interest in making the minimum any less than $15, according to a Democratic aide who declined to be named.

Republicans, who will still control the White House and Senate, appear to have little appetite for a federal minimum wage hike. A Wednesday morning hearing on a $15 minimum wage before the Education and Workforce Committee's Subcommittee on Workforce Protections will offer some clues into how the messaging battle will play out starting in January. The GOP still leads the committee until next month.

(The Wednesday hearing was canceled after a homophobic blog post written by a Republican witness, San Diego State University professor Joseph Sabia, surfaced. Sabia did not immediately respond to CNBC's request for comment).

Even if Democrats fail to raise the federal pay floor, they see the message as a winning one. The party has hammered the GOP as out of touch with workers and friendly to corporations. The minimum wage fight will offer ample opportunities to drill into that argument ahead of President Donald Trump's 2020 re-election bid.

"Next year, this Committee will work to build an economy that works for everyone, not just the wealthy few," Scott said in a statement Friday after the release of a solid November jobs report. He cited "raising the federal minimum wage" as one of the tools to reach that goal.

Republicans appear unlikely to get behind the policy. While Trump has voiced support for a higher minimum wage, he has typically said states should decide their own pay floors. A $15 minimum wage would also have a tough path in the Senate, where the GOP will control 53 of 100 seats next year. Powerful business groups including the U.S. Chamber of Commerce have also opposed the proposal.

Senate Majority Leader Mitch McConnell's office declined to comment on next year's legislative agenda. The White House did not respond to requests to comment.