WASHINGTON—Republicans blocked the Senate from voting Thursday on two of President Barack Obama's nominees, reviving the prospect of a broader fight over nominations that both parties have said could poison relations and shut down all other business in the chamber.

Two prominent Obama nominees—Rep. Mel Watt (D., N.C.), picked to head the agency that oversees Fannie Mae and Freddie Mac , and Patricia Millett, nominated for a seat on a key federal appeals court—failed to clear procedural hurdles in back-to-back votes. Each fell short of the 60 votes needed to advance.

The GOP opposition angered the majority Democrats, who said they were considering whether to take new steps to limit Republicans' ability to prevent confirmation votes.

The rules for confirming nominees are arcane but highly sensitive because they give the minority party the power to flex its muscles by requiring the majority to secure 60 votes to cut off debate before proceeding to a confirmation vote, which requires only a simple majority.

Thursday's votes—one of which involved the rare step of blocking a sitting member of Congress from an executive post—threatened to interrupt a period of relative comity in the two weeks since the two parties ended a budget fight that led to a 16-day partial government shutdown. Lawmakers earlier this week confirmed nominees to two key federal agencies and began budget negotiations Wednesday with an unusually conciliatory meeting.