One viewer wrote on Twitter: "Subtle of the BBC to show a clip of Savile introducing Cliff Richard on TOTP."
The threat of legal action follows the controversy surrounding the 2014 police raid on Sir Cliff's £2.5 million (NZ$5.2 million) Berkshire home, which officers let the BBC film after hatching a secret deal.
At the time, Sir Cliff's lawyers were considering suing the BBC for breach of privacy. A subsequent report concluded the BBC's actions had caused the singer "unnecessary distress".
Detectives have questioned Sir Cliff twice over historical sex assault allegations and have spent almost two years trying to gather evidence.
However, South Yorkshire Police has still not sent its case file to the Crown Prosecution Service, which will decide if the allegations against him are strong enough to press charges.
Sir Cliff is understood to have provided police with evidence that he was never alone with two of the men who allege he attacked them in the 1980s.
The singer, who completed a comeback tour to celebrate his 75th birthday last year, has denied the allegations.
The police are coming under increasing pressure to bring their inquiries to a conclusion, either by bringing charges against the star or by exonerating him.
The BBC declined to comment.