Mr Thompson, who was in charge when the Newsnight investigation into Savile was dropped, is due to start his new role as president and chief executive officer of the paper on November 12.
Margaret Sullivan, the paper's independent public editor, wrote :
"How likely is it that (Thompson) knew nothing?
"How likely is it that the Times Company will continue with its plan to bring Mr Thompson on as chief executive?
"His integrity and decision-making are bound to affect The Times and its journalism - profoundly.
"It’s worth considering now whether he is the right person for the job, given this turn of events."
Amid the fallout from the Jimmy Savile scandal the BBC is investigating nine allegations of sexual harassment among staff and contributors.
BBC boss George Entwistle has spent almost two hours taking questions from MPs about the corporation's handling of the Savile scandal.
Ex-BBC boss Mark Thompson has told ITV News he will help with any inquiry into the corporation and its handling of the Savile allegations.