The headhunters have been impressed by his record at the Bank of Scotland, which is widely regarded as a well-run organisation despite the dramatic rise in bad debts that held back the group's profits last week.
'He has shown himself to be a very able banker,' said one leading analyst. 'The only question mark would be how he can translate that from a relatively small-scale operation like the Bank of Scotland to the massive bureaucracy that is Barclays.'
Another quality highlighted by Spencer Stuart is Mr Burt's experience outside banking - he worked in the computer industry until 1985.
But it is understood that the 49-year-old Scot may not want the job. He does not want to leave his beautiful house in North Berwick and is believed to be unwilling to act as superior to his old friend David Band, who heads Barclays' investment banking side. Mr Burt declined to comment.
A lack of interest on his part could leave the door open for another name on Spencer Stuart's shortlist, John Tugwell, head of National Westminster's US operations. His experience in turning around a banking business that has suffered large losses on its loan book could meet Barclays' current needs.
He is also believed to be disappointed about being passed over for the job of chief executive of NatWest, which went to a younger man, Derek Wanless.
Spencer Stuart would not even confirm that it had been appointed, although Barclays admitted that it had hired the firm.
The headhunters have been told that their brief is to find a chief executive with wide experience, and it is thought they are also looking at candidates from outside the banking industry.
The firm was appointed last month after Andrew Buxton, Barclays' chairman and chief executive, bowed to City pressure to split the roles. But at the bank's annual shareholders' meeting he defined the jobs in a way that appeared to limit the role of the chief executive. He said that the chairman would be in charge of strategy and the chief executive would implement it.
A number of senior bankers expressed concern that the role of chief executive would be difficult to define at a bank that has never previously separated the functions and is only doing so reluctantly.
There are also worries about how other directors of Barclays would react to an outsider. Alistair Robinson, vice-chairman in charge of banking, has contemplated leaving over changes in the management structure, and there is a question mark over the future role of Sir Peter Middleton, the deputy chairman, once of the Treasury.
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