If confirmed, his appointment would make him one of the most prominent businessmen in Scotland after devolution. As chairman of giant brewer Scottish & Newcastle he employs 5,500 people north of the border.
The Bank of Scotland, which would make no comment on the weekend reports, is one of the UK's best-regarded financial institutions, whose profits climbed by pounds 119m to pounds 664m in the year to February. The note-issuing bank will become even more influential after devolution.
Yet, like the current governor, Sir Alistair is a staunch and high-profile opponent of a new Scottish parliament having tax-raising powers. The 60- year-old, who also recently stood down as head of the Safeway supermarket group, was criticised last month by Scottish Secretary Donald Dewar for warning that S&N could switch investment to England if business rates were increased in Scotland.
His stand echoes widespread concern in the Scottish business community, however.
Today sees the launch of a campaign by the British Retail Consortium lobbying against the devolution White Paper's proposal for business rates in Scotland to vary from the rest of the UK.
The BRC has written to Mr Dewar, warning him that fears of a higher business rate in Scotland would discourage future job creation and investment by retailers.
Sir Alistair, who has sponsored the world premiere in Edinburgh of a new Sir Peter Maxwell Davies composition on the Jacobite uprising, has been a staunch Conservative supporter. Before the election he said he would strongly back the Tories in Scotland, saying: "Some Tories need a kick up the bottom for their divisiveness and disloyalty."
However, this summer Scottish & Newcastle broke with the habit of 20 years by cancelling its pounds 50,000-a-year donation to the Conservative Party.
"The board agreed it was inappropriate to make any donation to any political party," Sir Alistair said.
The Bank of Scotland's new governor is due to be named on or before the announcement of its interim results on 1 October. Sir Alistair has served as a non-executive director on the Bank of Scotland board for five years.