The level of derision directed at Mr. Hagel from Republicans has been striking not just because defense secretaries are usually confirmed on a simple up-or-down vote, but also because Mr. Hagel, a Republican, served with many of them in the Senate until 2008.

“Senator Hagel’s performance at his confirmation hearing was deeply concerning, leading to serious doubts about his basic competence to meet the substantial demands of the office,” they said.

Senate Republicans narrowly blocked a vote on Mr. Hagel’s confirmation last week in a filibuster, forcing Democrats to put the matter off until senators return from recess next week.

Republicans have been using the filibuster to prevent final consideration of the nomination by refusing to end debate on it, a procedural step that requires 60 senators to vote in the affirmative.

But some Republicans, including Senator John McCain of Arizona, have since said that they will drop their objections. Mr. McCain was firm, saying on Sunday, “I don’t believe that we should hold up his nomination any further.”

Others, like Mr. Graham, Mr. Shelby and Ms. Fischer, have said that while they do not support a filibuster, they believe that the senators should have ample time to consider their votes, leaving themselves open to voting not to end debate next week.

Only one more Republican “yes” vote would be needed to cut off debate and carry through with a final vote if all the Republicans who voted to end the filibuster last week voted to do so again.