WASHINGTON—Sen. Dianne Feinstein has reported to the FBI an allegation concerning Supreme Court nominee Brett Kavanaugh that appeared to be connected to an incident of potential sexual abuse when he was a teenager, a person familiar with the matter said.

Mrs. Feinstein, the top Democrat on the Senate Judiciary Committee, said Thursday she had “received information from an individual concerning the nomination of Brett Kavanaugh to the Supreme Court” and referred it to investigative authorities.

The California senator said the person providing the information, which she didn’t specify, had requested confidentiality. Further details of the allegation weren’t immediately available.

Democrats on the Judiciary Committee met informally Wednesday night to discuss the letter, a Senate staffer said.

A person familiar with the matter confirmed Mrs. Feinstein’s letter was referred to the Federal Bureau of Investigation as the agency that conducts background investigations of Supreme Court nominees.

“Upon receipt of the information on the night of September 12, we included it as part of Judge Kavanaugh’s background file, as per the standard process,” an FBI spokeswoman said.

An FBI official said the bureau hadn’t opened a criminal investigation into the matter.

Judge Kavanaugh didn’t respond to requests to comment.

A spokesman for Judiciary Committee Chairman Sen. Chuck Grassley (R., Iowa) said Mr. Grassley was aware of Mrs. Feinstein’s referral but hadn’t seen the letter. The chairman didn’t plan to change the committee’s vote on Judge Kavanaugh’s nomination, which is scheduled for Sept. 20, the spokesman said.

White House spokeswoman Kerri Kupec criticized Mrs. Feinstein for raising the matter so late in the process. “Not until the eve of his confirmation has Sen. Feinstein or anyone raised the specter of new ‘information’ about him,” she said. “Throughout 25 years of public service, the Federal Bureau of Investigation has thoroughly and repeatedly vetted Judge Kavanaugh, dating back to 1993, for some of the most highly sensitive roles.”

At last week’s confirmation hearing, Sen. Mazie Hirono (D., Hawaii) asked Judge Kavanaugh if as a “legal adult” he had “ever made unwanted requests for sexual favors or committed any verbal or physical harassment or assault of a sexual nature.”

Judge Kavanaugh said no. To a follow-up question, he said that he had never faced discipline or entered into a settlement over such matters.

Ms. Hirono said she regularly asks judicial nominees whether they engaged in sexual harassment or assault “because it’s so hard to hold lifetime appointees to the federal bench accountable and because I did not want the #MeToo movement to be swept under the rug.”

A Hirono aide said that at the time the senator had no specific information regarding Judge Kavanaugh.

In another development, Judge Kavanaugh said in written answers to senators considering his nomination that he had previously spoken with President Trump’s transition team about becoming U.S. Solicitor General.

Judge Kavanaugh, who now serves on a federal appeals court, told the Senate Judiciary Committee that he discussed the solicitor general position with then-Sen. Jeff Sessions, the Alabama Republican who hadn’t yet been confirmed as attorney general, in a conversation “arranged by members of the presidential transition team.”

The conversation suggests the judge was at least willing to consider the possibility of serving as a political appointee in the Trump administration. The solicitor general represents the federal government in litigation before the Supreme Court. President Trump ultimately tapped lawyer Noel Francisco for the job.

“I was uncertain about it, but I was interested in learning more,” Judge Kavanaugh wrote. “I ultimately decided that I wanted to remain a judge.”

The judge’s comments were included in 263 pages of answers he submitted in response to written questions from the Judiciary Committee. Senators submitted the questions as a follow-up to last week’s confirmation hearings, where Judge Kavanaugh appeared for three days.

The committee on Thursday scheduled a Sept. 20 vote on Judge Kavanaugh. The judge is expected to be approved on a party-line vote, with Republicans in the majority. His nomination then would move to the Senate floor for final consideration.

The Supreme Court’s new term is set to begin Oct 1, and Senate Majority Leader Mitch McConnell (R., Ky.) has said he would like the full Senate to vote on Judge Kavanaugh by that date.

The nomination continues to be the subject of deep partisan disagreement. During the Judiciary Committee meeting Thursday, senators renewed their battle over access to documents from Judge Kavanaugh’s time as a lawyer in the George W. Bush White House.

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Republicans say ample public material is available to evaluate Judge Kavanaugh. Democrats say the GOP is rushing the confirmation process and precluding them from seeing additional documents that might shed light on the judge’s views about important issues that may come before the Supreme Court.

Committee Democrats on Thursday offered a series of motions seeking to subpoena additional White House documents, but Republicans voted down the requests.

In his written answers to the senators, Judge Kavanaugh adopted a cautious tone and, as he did during last week’s hearings, avoided taking positions on hot-button issues such as abortion. His answers were released Wednesday night.

The judge shed additional light on his personal finances, amid questions spurred by his financial disclosure forms as a judge that showed him carrying tens of thousands of dollars in debt. He said one debt that appeared on certain disclosure reports was a federal government loan to help with a down payment on his house, and other debts related to home improvements and repairs.

“Over the years, we have sunk a decent amount of money into our home for sometimes unanticipated repairs and improvements,” the judge wrote. “As many homeowners probably appreciate, the list sometimes seems to never end.”

Judge Kavanaugh said he and his wife now have no debts other than their mortgage. He also addressed questions about his expenditures on Washington Nationals baseball tickets, saying he bought four season tickets every season from 2005 through 2017, plus playoff tickets. “I am a huge sports fan,” he said.

The judge said several friends split the season tickets with him and reimbursed him for their portion of the costs.

—Sadie Gurman

and Brent Kendall contributed to this article.

Write to Kristina Peterson at kristina.peterson@wsj.com and Jess Bravin at jess.bravin@wsj.com