Asked by reporters if he would consider changing Senate rules if Millett was blocked, Reid said, "I'm not going to be talking about hypotheticals."
But New York Sen. Chuck Schumer, the No. 3 Senate Democratic leader, wouldn't rule it out in a brief interview, saying, "Have to see, have to see."
Millett, an attorney in private practice, has argued 32 cases in front of the Supreme Court and previously served as an assistant U.S. solicitor general under Presidents Bill Clinton, a Democrat, and George W. Bush, a Republican.
After months of Democratic accusations that Republicans were stalling Obama's efforts to fill key vacancies, the two parties reached a deal in July in which some GOP senators agreed to free several key nominees for votes. In exchange, Democrats agreed to drop a threatened effort to weaken the minority party's legislative powers.
As part of that deal, Obama removed Griffin as an NLRB board member, but he was to be given the general counsel slot, according to participants in that bargaining. The general counsel, who holds a four-year term, investigates and prosecutes cases before the board.
Republicans said this week that they were opposing Griffin, a Democrat and longtime labor lawyer, because the NLRB has become too pro-union. The agency's general counsel investigates and prosecutes cases before the board.
Watt seemed a longshot to win approval. Obama wants him to head the housing agency that oversees Fannie Mae and Freddie Mac, the huge government-controlled companies that own or guarantee about half of U.S. mortgages. They say the longtime congressional veteran lacks the technical expertise to head the agency and won't be politically impartial, charges Democrats deny.
The Senate planned votes on other nominations this week, some of which have seen GOP opposition fade. All were expected to succeed. They were:
—Alan Estevez for a top Pentagon procurement job.
—Katherine Archuleta to lead the Office of Personnel Management, which helps oversee federal workers.
—Jacob Lew, the treasury secretary, to represent the U.S. at several international financial organizations.
Associated Press writer Henry Jackson contributed to this report.