The bank's chief executive, Peter Burt, will inform Mr Robertson of the decision when the two men meet in the United States later today. Mr Burt flew to the United States on Wednesday after about pounds 400m was wiped of the value of the bank's shares amid the furore surrounding Mr Robertson's remarks.
Bank officials came to the decision after watching a video of Mr Robertson's remarks about Scotland in which he said: "You can't believe how strong the homosexuals are." The hugely successful right-wing preacher, who once sought the Republican candidacy for president, described Scotland as a nation in decline which "could go right back into darkness very easily".
Last night the Bank of Scotland could not officially confirm its decision, apparently reluctant to break the partnership publicly before meeting Mr Robertson. However, it is understood Mr Robertson's remarks have given officials a way out of a deal that for weeks has been a public relations disaster.
The bank had tried to weather the storm around its telephone banking deal with Mr Robertson. It had acknowledged that up to 500 people have switched their accounts to other banks. However, reports about his comments on Scotland proved to be a breaking point. The bank risked losing large institutional investors and was faced with little choice other than to pull the plug on the deal.
Yesterday, amid speculation that the deal would be called off, the bank's shares closed slightly up.
Laura Lambie, analyst with Capel Cure and Sharp, said; "There has been more and more bad publicity. We've had protest groups picketing the bank. Now the market has looked at the whole deal and thought, is this a good idea, should the bank be continuing with it and with bad publicity."
Mr Robertson made his remarks on his Virginia-based Christian Broadcasting Network and his 700 Club television show. Yesterday, a spokeswoman for the network could neither confirm nor deny the deal had been abandoned.
However, Gene Kapp, a spokesman for Mr Robertson, had tried to save the deal with conciliatory remarks. He said Mr Robertson's remarks had been taken out of context.
"He indicated that Scotland has a great, proud history and like many places in Europe and in the United States, what really needs to happen is a return to traditional values, period.
"What has happened is there has been, in some of the Scottish and UK media, an effort to distort his position on issues and there has been a republication of the quotes attributed to him that are either not factually correct or taken out of context in many cases."
The Bank of Scotland clearly became particularly worried about carrying on with the deal when debate began among members of the new Scottish Parliament on whether they should close their accounts with the bank.
West Lothian Council was due to meet later this month to consider withdrawal of funds. The Scottish Trades Union Congress, which holds far more sway than the TUC in England, also threatened to withdraw the business of 100,000 members who hold the Bank of Scotland Affinity card.
The scrapping of the link with Mr Robertson means that the bank has now lost a route into the lucrative US financial market. The bank had hoped that Mr Robertson's access to a database of hundreds of thousands of Christians would provide it with a plentiful supply of customers. However, such access had never been guaranteed by Mr Robertson.
In any case, the bank's problems in breaking into America are nothing compared with what might have resulted from continuing the partnership with the controversial evangelist.Reuse content