WASHINGTON — A fresh feud over federal judgeships has again begun to agitate the Senate, with Republicans so far blocking President Obama from filling any of the four vacancies on the nation’s most prestigious and important appeals court.

After Republicans this week filibustered the nomination of Caitlin J. Halligan of New York to the United States Court of Appeals for the District of Columbia Circuit, Democrats quickly began discussions over how to respond. One possibility is for Mr. Obama to make several simultaneous nominations, in effect daring Republicans to find specific objections in multiple instances. Democrats say Republicans would be hard pressed to come up with legitimate reasons to disqualify all four.

“We need to design a strategy to counter the Republicans, and we are going to need the president,” said Senator Tom Udall, Democrat of New Mexico. “Rather than putting just one up, we should put before the Senate all four and expose what is happening here.”

If Republicans were to continue to steadfastly block a series of appeals court nominees, Democrats say they might then have justification to revisit Senate rules and claim new power to thwart filibusters.