Updated

Several senators are urging the Senate Foreign Relations Committee to bring in Secretary of State Hillary Clinton to testify on the Libya terror attack in an open hearing "as soon as possible," expressing concern that months have passed without her testimony.

Committee Chairman John Kerry, D-Mass., on Friday announced that Clinton has agreed to testify before the end of the current session, noting that the State Department's internal review is nearly complete.

But amid claims that the hearing could be closed, the lawmakers who wrote to Kerry the same day made clear that they want her testimony to be public -- and they don't want it held up much longer.

"Nearly three months have passed since the terrorist attacks in Benghazi, yet we still have far more questions than answers about conditions in Benghazi prior to the attacks, the situation on the ground that night, and the administration's handling of its response to the attacks," they wrote.

"We recognize that the State Department's Accountability Review Board investigation is a useful tool, but it is no substitute for our own investigation into the circumstances surrounding the attacks."

The lawmakers included Sen. John Barrasso, R-Wyo., and Jim DeMint, R-S.C. They called for hearings to be scheduled this month.

Tuesday marks three months since the deadly attack in which four Americans were killed. Those three months have been punctuated by conflicting accounts about what warnings the administration may have been given in the run-up to the attack. Further, lawmakers continue to quarrel about U.S. Ambassador to the U.N. Susan Rice's initial claim that the attack was effectively a protest spun out of control.

The administration has since walked back that claim, but questions remain about the true nature of the attack and why Rice initially used that characterization.

Clinton is expected to testify before two committees, with the other being the House Foreign Affairs Committee. Rep. Ileana Ros-Lehtinen, chairwoman of that committee, told Fox News on Saturday that Clinton's office confirmed she will testify on the House side in an open setting.