Others in the state said there is little that can be done, as the December 12 election to fill the seat vacated by Sessions approaches.
Speaking at a House Judiciary Committee hearing Tuesday, Sessions said he had "no reason to doubt" the women who have made the allegations against Moore.
However, Moore has flatly denied the allegation levelled against him.
"We will evaluate every case as to whether or not it should be investigated", Sessions replied.
He also said that if Moore does not step aside, the Senate will "need to act" to protect its integrity.
Mr. McConnell says he intends to discuss Mr. Moore's situation with President Trump, who returns on Tuesday evening from his 12-day whirlwind tour of Asia.
The Alabama Republican Party and Moore's official senatorial campaign are now the only two entities listed on the joint fundraising agreement.
"As of today, with the information that's been introduced to me, and if these charges are not proven to be true, then I would continue to support and vote for Judge Moore", said Alabama Secretary of State John Merrill, R, in a Tuesday interview with CNN.
The national Democratic Party, however, has yet to directly invest in the Alabama race beyond funds to build the state party, and has not changed that stance since the Moore allegations surfaced. And on Monday, a separate Alabama woman alleged Moore sexually assaulted her when she was a teenager, and described her experience at a news conference, represented by attorney Gloria Allred.
"Do you believe Roy Moore over the women?" the reporter asked.
Moore, a Christian conservative and former chief justice of the Alabama Supreme Court, said on Monday that Nelson's accusations are "absolutely false".
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