The rule change they would seek is intended to be limited. It would allow senators to continue to filibuster legislation and judges, but not appointments to federal agencies or cabinet posts.

Democrats believe that their argument — that a president has the right to assemble his own team of like-minded cabinet officials and other high-level policy makers — is more persuasive in the court of public opinion. They also believe that this fight could have fewer consequences for them should their political fortunes reverse and they find themselves in the minority trying to block judicial nominees from a Republican White House.

Democrats are still strategizing over how best to proceed, but the nominees they have talked about putting forward first are those for vacant seats on the National Labor Relations Board, the government entity that has become a major source of contention in the fight over confirmations between the White House and Senate Republicans.

The others are also divisive: Richard Cordray as director of the Consumer Financial Protection Bureau, Thomas E. Perez as secretary of labor and Gina McCarthy as director of the E.P.A.

“What’s particularly galling to us is there are certain vacancies that haven’t been filled not because the Republicans have anything against the nominee,” said Senator Charles E. Schumer of New York, the No. 3 Senate Democrat. “But rather because they just dislike the agencies and they don’t want them to function.”

Republicans contend that Mr. Obama has been too heavy-handed in making some of his appointments, which have drawn scrutiny by the courts. Mr. Cordray and three of the five National Labor Relations Board nominees are, in fact, already serving because the president installed them using a backdoor maneuver known as a recess appointment, allowing him to bypass the Senate. The Supreme Court has agreed to decide whether Mr. Obama violated the Constitution last year when he made the three labor board recess appointments. Mr. McConnell has called those appointments an “unprecedented power grab.”

Because of the doubt the courts have cast on the legitimacy of the labor board appointments, the legality of the board’s decisions in the year and a half since Mr. Obama went around the Senate and named them is in question. So to clear up any legal ambiguities, the president has asked the Senate to confirm all five members of the board, which rules on matters like disputed collective bargaining agreements.