The magazine identified the woman as Deborah Ramirez (53).
Ramirez alleges that Kavanaugh exposed himself to her at a college party, where they had both been drinking, "thrust his penis in her face, and caused her to touch it without her consent as she pushed him away".
On Wednesday, Trump called the allegations against Kavanaugh "highly unsubstantiated", and said "you should look into the lawyers" representing the accusers.
White House spokesperson Kerri Kupec told Fox News: "This 35-year-old, uncorroborated claim is the latest in a coordinated smear campaign by the Democrats created to tear down a good man".
Ford has agreed to testify before the Senate Judiciary Committee, and has asked that the FBI investigate her claim, and yet, Republican Senators have casted doubt on her account, stating they will advance Kavanaugh's nomination proceedings regardless of her allegations.
The judge has denied both allegations, labelling the latest "a smear". The effect of believing accuser Christine Blasey Ford - who maintains that a 17-year-old intoxicated Kavanaugh pinned her to a bed at a high school party, groped her, put his hand over her mouth, and attempted to remove her clothes - is not that Kavanaugh would lose his liberty and go to jail.
"Following Dr Ford's testimony, Judge Kavanaugh will appear again before the committee", a media statement said.
Accuser of Supreme Court nominee Kavanaugh agrees to testify to Senate committee
Ford committed to an "open" hearing after negotiators convened a call on Sunday, her attorneys said in a statement . Ford's legal team has said she strongly prefers that her allegations are fully investigated before she testifies.
The committee said it would not negotiate on other points, though, including Ford's desire for additional witnesses and a request to testify after, not before, Kavanaugh.
In addition, Katz is the vice-chair of the Project on Government Oversight, a group that wrote to Feinstein and Senate Judiciary Committee chair Chuck Grassley (R-Iowa) demanding the committee stall Kavanaugh's nomination until further records could be released. I then answered more questions at a confidential session.
Ms Ford's lawyers said it was still unclear who will ask questions, as Republicans were trying to hire an outside female counsel who could take over the questioning.
Trump also doubled down on his belief that the sexual assault allegations against Kavanaugh are suspect as they were not reported earlier.
After going over the timeline of events leading up to the leaking of the story about a week ago, he asked "Does this sound like Democratic senators take their responsibility seriously and want to get to the truth?"
In a statement, Grassley said the Judiciary committee's majority staff learned of the allegation made by Ramirez from The New Yorker's report. On Sunday, Sen. Lindsey Graham admitted that he planned to vote for Kavanaugh regardless of what Ford said in her testimony.
"If they continue to contest those two things, there won't be a hearing", Mr Graham said. "Let's figure it out", she urged. "It was a story that was known, but it was a story that i was embarrassed about", she admitted.
Michael Avenatti, a USA lawyer known for representing adult film actress Stormy Daniels in her cases against President Trump, has alleged on Twitter that he is representing a third woman with "credible information" regarding Judge Kavanaugh and the alleged witness to Prof Ford's assault, Mark Judge.