He told Richard Bacon's Five Live Show that he was still enjoying presenting the BBC2 current affairs flagship show but admitted it had been damaged.
Asked if he considered quitting, he said: "I thought about it of course but in the end I decided that you know...there were several bad decisions, they were individual bad decisions and I felt that loyalty commanded that you stayed or I stay."
He also said senior figures proposed dropping the show.
He said: "It is my belief, I have never checked this out, that there were a couple of people on the BBC Trust who thought, who over-reacted, who thought that it should be shut down."
But Paxman said that was "never a serious possibility" and he would stay with the show "as long as somebody asks me".
The decision not to run the Savile story, which was picked up by ITV and sparked a major police investigation, indirectly cost director-general George Entwistle his job.
A review of events revealed rivalries and factional in-fighting at the BBC and said the decision to drop the Savile report was "flawed".