Judge Kavanaugh and Christine Blasey Ford, the woman who says he attacked her more than three decades ago, will both be able to testify.
US President Donald Trump said earlier he expected "a little delay" to confirming his nominee.
Judge Kavanaugh, 53, says the allegation is "completely false".
He denies he was even at the 1982 high school party in question where his accuser, now a psychology professor in California, says he tried to rape her as his friend watched.
The claim has jeopardised Judge Kavanaugh's formerly all-but certain-nomination for a lifetime job on the Supreme Court.
What did the committee say?
Senate Judiciary Committee Chairman Chuck Grassley confirmed next Monday's hearing in a statement.
"As I said earlier, anyone who comes forward as Dr Ford has done deserves to be heard," said the Iowa Republican.
Mr Grassley had earlier stopped short of calling for a public hearing, or for delaying the committee's vote on the nominee.
The judge last week finished four days of hearings before the Senate Judiciary Committee, and a vote on his confirmation had been planned for Thursday of this week.
But Democrats demanded a delay in the vote in order to let the FBI investigate.
Mr Grassley said the standard procedure would be for committee members to conduct telephone calls with both witnesses about their forthcoming testimony.
But in a statement, the panel's Democrats refused to join in any phone call with Judge Kavanaugh.
What are other senators saying?
Senators from both parties had argued that Judge Kavanaugh and Prof Ford should be given the chance to testify before the panel.
Senator Susan Collins, a Maine Republican, told reporters she would like the chance to observe Prof Ford to decide the credibility of her account.
"Obviously, if Judge Kavanaugh has lied about what happened, that would be disqualifying," she said.
Two other Republican senators, Senator Jeff Flake of Arizona, and Bob Corker of Tennessee, earlier called for the vote to be delayed.
What is the allegation?
Prof Ford detailed her story in a letter sent to Senate Judiciary Committee senior Democrat Dianne Feinstein in July before going public in an article published in the Washington Post on Sunday.
The Palo Alto University academic says the incident happened 36 years ago in a Maryland suburb outside Washington DC.
Prof Ford says Brett Kavanaugh pinned her to a bed and tried to take off her clothes, then clamped his hand over her mouth when she attempted to scream.
They were students in different high schools. She was 15 and he was 17.
Both he and his friend, who allegedly watched the incident, were drunk, she says.
Mark Judge, who has been named as the third person in the room, told The Weekly Standard the allegations were "absolutely nuts".
Prof Ford says she spoke of the alleged attack a couple of times over the subsequent decades, including during therapy.
In a statement on Monday, Judge Kavanaugh vowed to "refute" the accusation before the congressional panel.
He said in a statement issued by the White House: "I have never done anything like what the accuser describes - to her or to anyone.
"Because this never happened, I had no idea who was making this accusation until she identified herself yesterday."
Prof Ford's lawyer, Debra Katz, told NBC's Today programme on Monday that "if it were not for the severe intoxication of Brett Kavanaugh, she [Prof Ford] would have been raped".
What's at stake?
Republicans control the Senate by only a narrow 51-49 margin, meaning any defections could scupper the nomination.
That would set back President Trump's efforts to install more conservatives on the Supreme Court and broader US judiciary.
If confirmed, Judge Kavanaugh, a conservative federal appeals court judge, would be expected to tilt the court's balance further to the right.
The confirmation fight comes just weeks before November's congressional elections in which Democrats are hoping to wrest control of Congress from Mr Trump's party.
The Judicial Crisis Network, a conservative group that backs Mr Trump's judicial picks, says it plans to launch a $1.5m (£1.1m) ad campaign backing Judge Kavanaugh.
What does the White House say?
Mr Trump, who faces multiple accusations of sexual misconduct that surfaced during the 2016 presidential election campaign, has not offered an opinion on the credibility of Prof Ford's allegation.
At the White House on Monday, the Republican president said of the confirmation: "If it takes a little delay, it will take a little delay."
He dismissed as "ridiculous" a reporter's question about whether Judge Kavanaugh's nomination would be withdrawn.
"I think he's very much on track," Mr Trump said.
The White House said in a statement: "Judge Kavanaugh looks forward to a hearing where he can clear his name of this false allegation."
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