The Dean of Durham, the Very Rev John Arnold, also reopened the debate on the bishop's future by claiming he had been considering resigning since the conviction was revealed by a Sunday newspaper last week.
Interviewed on BBC Radio 4's Sunday programme, the dean was asked if the bishop was considering resignation.
He replied: 'Obviously he has been considering it. The thought has been going through his mind.'
Last night, the dean said that he did not think that the bishop had ruled out resigning. But a spokesman for the Rt Rev Turnbull said he would not resign and was looking forward to starting his new job on 22 October.
The dean's comments were the first suggestion that the bishop might be contemplating standing down from his post, which he took over from Dr David Jenkins.
The bishop, who is married with three grown-up daughters, was convicted of an act of gross indecency with a Yorkshire farmer in 1968. Speaking of the bishop's ordeal, the dean said: 'He's been having a very rough time.'