"My view is we should hold on this until we get sufficient information," said Ayotte, adding she "would place a hold on anybody who wanted to be promoted for any job who had a role in the Benghazi situation."
The New Hampshire Republican's words were echoed by fellow GOP Senate Armed Services Committee member Lindsey Graham of South Carolina.
"Before anyone can make an intelligent decision about promoting someone involved in Benghazi, we need to do a lot more," Graham said. "We're not even close to getting the basic answers."
Ayotte, Graham and Senate Armed Services Committee member John McCain, R-Ariz., met with Rice, the U.S. ambassador to the United Nations, and Michael Morell, acting CIA director, behind closed doors Tuesday.
Rice and Morell are reported to be under front-runner consideration to serve as the next secretary of state and CIA director, respectively.
The two later met with Sen. Joseph Lieberman, Ind-Conn., whose Homeland Security Committee is also probing the Sept. 11 attack that killed U.S. Ambassador Christopher Stevens and three other Americans.
Rice and Morell were to meet privately Wednesday with Sen. Susan Collins, R-Maine, the Homeland Security panel's second-most-senior member.
"We are significantly troubled by many of the answers that we got, and some that we didn't get," McCain told reporters.
White House press secretary Jay Carney rejected the Republicans' charge Rice and Morell left questions unanswered.
"I would simply say that there are no unanswered questions about Ambassador Rice's appearances on Sunday shows [Sept. 16] and the talking points that she used for those appearances that were provided by the intelligence community," Carney said.
Rice appeared on five interview shows that day -- five days after the attack -- saying the bombardment of the consulate began as a "spontaneous reaction" to an anti-Islamic video widely disseminated throughout the Arab and Muslim world that had also set off protests elsewhere.
That claim was later debunked by intelligence officials and reports from the ground.
"The questions that remain to be answered, and that the president insists are answered, have to do with what happened in Benghazi, who was responsible for the deaths of four Americans including our ambassador, and what steps we need to take to ensure that something like that does not happen again," Carney said.
In a statement issued after the meeting, Rice conceded her TV comments were incorrect but said they were based on evolving intelligence.
"The talking points provided by the intelligence community, and the initial assessment upon which they were based, were incorrect in a key respect: There was no protest or demonstration in Benghazi," she said in her statement.
Rice said she didn't intend to mislead and said "the administration updated Congress and the American people as our assessments evolved."
Lieberman said after his meeting with Rice she "told the truth to the best of her knowledge" Sept. 16, The Washington Post reported.
"I asked if she was briefed by the White House, the campaign, or the political operation, and she said she had seen no message points from the White House," the Post quoted Lieberman as saying. "Nothing she said [on the Sunday shows] disqualifies her for any other office."
Lieberman will not participate in the Senate confirmation hearings. He retires in January.