“The fact is, no one knows who is going to win the election,” said Richard G. Lugar, the former Republican senator from Indiana. “Therefore, speculating that whoever is nominated is going to be in a better position may be an error.”

Mr. Lugar, a centrist who lost his re-election primary to a Tea Party-aligned candidate in 2012, urged his former Republican colleagues to move ahead once the president has made his choice.

“I can understand their reluctance given the controversy that surrounds all of the debate that has already occurred,” Mr. Lugar said. “But that is not sufficient reason to forgo your duty.”

His view was shared by Olympia J. Snowe, a former moderate Republican senator from Maine. “I believe that the process should go forward and be given a good-faith effort — and ultimately people will come to their own decision on a vote on a nominee,” she said in a statement.

Of course, Ms. Snowe and Mr. Lugar no longer have a vote in the Senate. Many would point to their very absence — and the continuing decline in the number of lawmakers like them — as a major contributing factor to the gridlock that has plagued Congress, leaving so few voices in the middle to provide a counterweight to the extremes on both the right and left.

The incipient Supreme Court fight moves into a new phase this week as lawmakers return to Washington after a holiday recess that followed Justice Scalia’s death. Now they will take their perspectives on the court fight to the committee rooms and hallways of the Capitol, as well as to the floor of the Senate and even the House, where conservatives will exhort their Senate colleagues to stand strong while Democrats denounce the Republicans.

Senator Mitch McConnell, the Kentucky Republican and majority leader, seems determined to hold his ground against initiating confirmation proceedings. Democrats, heartened by polls that show most Americans believe the Senate should act to fill the vacancy, intend to try to make Republicans pay for their refusal to do so. The clash is likely to erase even the modest gains that have been made in making the Senate more productive as each side seeks opportunities to punish and embarrass the other.