The council has stopped short of closing its account with the bank. But after considering the issue in private at its quarterly meeting in Edinburgh on Wednesday Ivor Sutherland, the registrar, made clear it remained "uneasy and uncomfortable".
Mr Sutherland said: "We will not come to any over-hasty decision but we will continue to put pressure on the bank and we will look at the matter again in October."
As it happens, the GTC held its discussion on the day the Bank of Scotland announced it was reviewing its relationship with Mr Robertson, following a video broadcast on the right-wing preacher's Christian network in which Scotland was described as a "rather dark land" which tolerated homosexuals.
The bank earlier told the GTC it had met the Church of Scotland which said there was no evidence that would lead the Kirk to close accounts or sell its shares.