Sir Alistair, 62, is understood to have been suffering from cancer and has asked for time off to complete his recovery after a series of operations.
His job at Bank of Scotland will be filled for the time being by his deputy Sir John Shaw, while Brian Stewart, the chief executive and deputy chairman, will stand in at Scottish & Newcastle.
Mr Stewart said yesterday that he and his colleagues on the board were "looking forward to Sir Alistair making a speedy recovery".
Peter Burt, the chief executive, was travelling yesterday and was unavailable for comment. However, in a statement yesterday the bank said: "Sir Alistair Grant has decided that, in order to assist his complete recovery from recent ill health, he wishes to step down as Governor of the Bank of Scotland for a period."
Sir Alistair, the former chairman of the supermarkets group Safeway, had been chairman at the bank for barely a year, having stepped up to the post only last May following the retirement of his predecessor, Sir Bruce Pattullo.
His appointment had been expected to signal a change in style for what has traditionally been a very conservative bank. However, because of his illness, he has been unable to devote himself as fully to the job as he might have wished.
Bank of Scotland insisted yesterday that the intention was for Sir Alistair to return to the chairmanship at a later date. To this end, he will stay on as a director in order to maintain his involvement with the bank's affairs.
However, he will not be seeking formal re-election as Governor, as the Bank of Scotland chairmen are known, at the bank's annual meeting on 15 June. He has been on the Bank of Scotland's board since 1992.
Bank of Scotland had hoped to keep Sir Alistair's illness out of the public eye. However, it is understood that the bank was forced to make a statement yesterday clarifying the position because of the need to notify shareholders in advance of the motions to be voted on at the annual meeting.