LONDON — Speculation about a political figure from the Margaret Thatcher era alleged to be tied to Britain’s child abuse scandal risks becoming a witch hunt, Prime Minister David Cameron warned Thursday after he was confronted in a television interview with names of supposed potential suspects.

Cameron has authorized two new inquiries into widespread child abuse at care homes, foster homes, and other institutions for vulnerable young people in Wales in the 1970s and 1980s. One victim has alleged that previous investigations had failed to examine accusations against a senior figure within Cameron’s Conservative Party at the time.