WASHINGTON—The nominee to run the Central Intelligence Agency appears to have the votes needed to be confirmed later this week, after winning over a senior Democratic senator who pressed her on the agency’s controversial interrogation program and picking up additional support from his side of the aisle.

Sen. Mark Warner of Virginia, the top Democrat on the Senate Intelligence Committee and an influential voice on national security, announced Tuesday he would back Gina Haspel. He cited a new letter in which she said that the agency shouldn’t have undertaken a post-9/11 antiterrorism program that many critics say amounted to torture.

Shortly after Mr. Warner’s statement, two more Democrats, Sen. Heidi Heitkamp of North Dakota and Sen. Bill Nelson of Florida, also announced they would support the nominee. Both are running for re-election in states that Republican President Donald Trump won in 2016.

Their support brings the number of Democratic senators backing Ms. Haspel to five, giving her additional breathing room in the Senate, where she had only a narrow margin until Tuesday. Most Republicans were expected to support her, while most Democrats were expected to oppose her. She needs a majority to be confirmed.

On the Republican side, Sen. John McCain, who was tortured while held prisoner during the Vietnam War, opposed her nomination over the interrogation program and called on the Senate to reject her. Mr. McCain is in Arizona undergoing treatment for cancer and is unlikely to return to Washington to vote. Sen. Rand Paul (R., Ky.) has said he would oppose Ms. Haspel. Among Republicans who had been publicly noncommittal, Sen. Mike Lee of Utah said late Tuesday he would back Ms. Haspel.