The fresh approach from Barclays is understood to have come within the last few weeks. It comes in the wake of the decision by Barclays' first choice, the American banker Mike O'Neill, not to take up the post because of ill-health.
However, Mr Burt who earlier this week faced flak from shareholders over the handling of the bank's abortive joint venture with Pat Robertson, the American preacher, has told Barclays that he would only take the job if the English bank agreed to a reverse takeover with the Bank of Scotland - in effect the same kind of deal already proposed to Barclays by Mr Burt's rival, Sir George Mathewson, at the Royal Bank of Scotland.
This would involve moving the Barclays group headquarters to Edinburgh and giving the Scottish bank a disproportionate share of the top jobs at the combined bank.
The proposal, perhaps unsurprisingly, has not met with approval from the Barclays board. Barclays had originally approached Mr Burt about the chief executive job earlier this year before settling on Mr O'Neill.
The fact that Mr Burt has not leapt at the offer illustrates the difficulty Barclays has been facing in trying to fill both the chief executive and finance director positions on the board. This is despite the reassuring noises Barclays has been making about progress of the search.
Barclays claimed when Mike O'Neill quit in March that having already narrowed its choices down to a very small shortlist it had a head start when it was forced to resume its search. However, it is now nine weeks since Mr O'Neill went and there is little sign that the bank is any closer to finding a replacement.
Other names that have surfaced recently include Malcolm Williamson, the former Standard Chartered chief executive, who now heads up Visa International in the United States. Mr Willamson was a former Barclays man before going to Standard Chartered and may be looking to return to the UK.
The bank is also looking again at John Varley, the head of its retail bank and the leading internal candidate.
So far Barclays shareholders have reacted relatively calmly to the long hiatus, helped by the favourable business climate and the firm leadership of Sir Peter Middleton, the Barclays chairman who stepped into the breach late last year.
On Wednesday, Barclays put out a bullish pre-close season statement confirming that plans to hold costs level this year were on track and that lending was up and bad debts down.
The bank's shares rose again yesterday.
However, analysts warn that with the Royal Bank of Scotland snapping at its heels, investor patience may start to wear thin if the top job is still unfilled in a few months time.