Sarah Huckabee Sanders, the White House press secretary, said Mr. Trump did not meet with any other candidates over the weekend. With less than a week before Mr. Trump’s self-imposed deadline of next Monday to announce his choice to succeed Justice Kennedy, who announced his resignation on Wednesday, the White House has embarked on a vetting and public relations effort that in past administrations has taken weeks.

But Ms. Sanders said that teams of lawyers from the counsel’s office and the Justice Department have begun compiling volumes of research on the finalists, and that key staff members would spend the summer focused entirely on winning confirmation of whomever Mr. Trump eventually chooses. And Democrats have responded to the accelerated White House effort by immediately targeting judges widely believed to be on the president’s shortlist.

The candidates interviewed Monday are among a group of federal appeals court judges who are believed to be finalists to replace Justice Kennedy. All are on a broader list of 25 people, mostly conservative judges, from which Mr. Trump has publicly said he will choose. His shortlist appears to also include Thomas M. Hardiman of the Third Circuit, William H. Pryor Jr. of the 11th Circuit and Joan L. Larsen of the Sixth Circuit.

Even without a name, Democrats began focusing on Monday on the backgrounds and rulings of the finalists, stressing their likely opposition to the Affordable Care Act and predicting that any of them would support overturning Roe v. Wade, the landmark 1973 case establishing the constitutional right to an abortion.

Senator Chuck Schumer of New York, the Democratic leader, took direct aim at Ms. Barrett on Twitter, accusing her of supporting the idea of rearguing the abortion precedent.

“The bottom line: Judge Barrett has given every indication that she will be an activist judge on the Court,” Mr. Schumer wrote in one of a series of tweets. “If chosen as the nominee, she will be the deciding vote to overturn Roe v. Wade and to strike down pre-existing conditions protections in the ACA.”

Mr. Schumer and other Democrats have insisted that Mr. Trump’s choice for the court must be pressed during hearings to specifically say whether they would join a majority in the court to end abortion rights.