That sets up a potentially contentious several weeks as Democrats continue to pressure Acting Attorney General Matthew Whitaker to recuse himself from the Mueller inquiry. Mr. Whitaker, who was publicly critical of the investigation before coming to the Justice Department, has so far signaled he will not step aside.

The concerns over the Mueller probe join broader questions about whether Mr. Whitaker, whom President Trump appointed after ousting Jeff Sessions on Nov. 7, can serve without Senate confirmation. Democrats in recent days have called for Mr. Whitaker to testify before Congress, sought a Justice Department inspector general’s investigation into his communications with the White House, and asked a federal judge to bar him from serving.

Democrats will gain some power when they take control of the House in January, but they say their most immediate leverage comes from Congress’s need to pass spending bills by Dec. 8. Republicans in both chambers say they want to avoid a partial government shutdown, and they need Democratic votes to do so.

Spending bills need 60 votes to clear procedural hurdles in the Senate, where Republicans currently hold 51 seats. The GOP tentatively gained two seats in the recent election, but the newcomers won’t take office until January.

If Congress and the White House cannot agree on a spending bill by Dec. 8, many agencies—including such high-profile entities as the Department of Homeland Security—will shut down. Democrats’ demand to protect the special counsel is just one piece of a complex fight over the spending package; Mr. Trump, for example, has said it must include $5 billion more for construction of a wall along the border with Mexico.