Alaska Sen. Lisa Murkowski said Monday she believes Brett Kavanaugh and the woman who has accused him of a decades-old sexual assault should both testify under oath to the Senate Judiciary Committee.

"Allegations surrounding sexual assault must be taken seriously and the Judiciary Committee must look into this further," Murkowski said in an e-mail through her press office Monday afternoon. On Sunday, Christine Blasey Ford, a research psychologist in Northern California, told The Washington Post that Kavanaugh had sexually assaulted her more than three decades ago when he was in high school.

"Despite the length of time since the alleged incident, Dr. Ford's allegations should be heard and she must have an opportunity to present her story before the committee under oath, with Judge Kavanaugh having the opportunity to respond under oath as well," Murkowski said.

Murkowski, R-Alaska, had previously signaled that she wanted to see a planned Thursday vote to advance the nomination of Brett Kavanaugh to the U.S. Supreme Court delayed while the allegation is investigated.

"If there is real substance to this, it demands a response," she told CNN on Monday.

Murkowski is considered a crucial Republican swing vote and has faced intense pressure from both liberal and conservative groups over her decision.

Sen. Dan Sullivan, R-Alaska, who has made combating domestic abuse and sexual assault a major policy focus, has publicly stated his support for Kavanaugh. But, he said Monday in a statement from his press office that "allegations of sexual assault should be taken seriously" and he wanted both the accuser and the accused to have a chance to speak.

"Both Professor Ford and Judge Kavanaugh deserve the right to be heard on this matter," he said. "I commend Chairman (Sen. Charles) Grassley (R-Iowa) on committing to as much and will be following closely as these claims are evaluated and reviewed."