At one point, the White House had hopes of winning the support of several Democratic senators running for re-election in states that Mr. Trump won, but that prospect seemed to vanish with the emergence of Dr. Blasey’s allegations. If no Democrats support Judge Kavanaugh, that means the Republicans, who hold a 51-to-49 majority with Vice President Mike Pence able to break ties, can afford to lose only one of their own caucus.

During the interview, Mr. Trump declined to say whether he might fire Rod J. Rosenstein, the deputy attorney general, after a report in The New York Times that Mr. Rosenstein last year suggested secretly recording the president to demonstrate the chaos in the administration and raised the possibility of invoking the 25th Amendment to remove him from office.

The Justice Department has said that Mr. Rosenstein was joking when he talked about taping the president, a version in conflict with others who described his comments. Mr. Rosenstein said that he currently sees “no basis to invoke the 25th Amendment.”

Advisers have sought to keep Mr. Trump from firing Mr. Rosenstein, at least until after the midterm elections in November, for fear of a backlash. Since Attorney General Jeff Sessions has recused himself from overseeing the Russia investigation, Mr. Rosenstein supervises the special counsel, Robert S. Mueller III.

“I don’t want to comment on it until I get all the facts,” Mr. Trump said when asked if he would fire Mr. Rosenstein. “I haven’t gotten all the facts, but certainly it’s being looked at in terms of what took place. If anything took place and I’ll make a determination sometime later, but I don’t have the facts.”

Mr. Trump asserted that he was not worried about his former personal attorney, Michael D. Cohen, who is talking with Mr. Mueller’s prosecutors after pleading guilty to breaking campaign finance laws and other crimes.

“No, I’m not, because I’m an honest guy,” Mr. Trump said. “And that stuff is unrelated to me. And that was Michael did that stuff. And Michael I mean you know, you know I have many, many lawyers. I have very big law firms I have that do my work.”