Disillusioned with the bank's poor performance, the institutions are anxious to split the roles of chairman and chief executive, which have traditionally been combined at Barclays. Shareholders lay some of the blame for Barclays' pounds 30m first-half loss - only the second loss in the bank's 300-year history - on an outdated board structure.
Sir John Quinton, the outgoing chairman, handed over the chief executive role to Mr Buxton at the Barclays annual meeting in April. At the same time, Mr Buxton was declared chairman-designate, but he does not assume the position until Sir John leaves at the end of this year.
One other candidate for the chairmanship is Sir Peter Middleton, a former permanent under-secretary at the Treasury. He is already executive deputy chairman of the group and head of BZW, the securities arm. Aged 58, he is four years younger than Sir John but older than Mr Buxton, who is 53.
The other contender is Sir Martin Jacomb, also a deputy chairman. He is handicapped by being the same age as the outgoing chairman. Insiders insist, however, that he has strong support among large institutional shareholders such as the Prudential and PosTel. His background in merchant banking - he was head of BZW when it was formed in the mid-1980s - means he has close contacts with City institutions.
There has been a move in British business over the past few years to divide the top roles in companies to avoid too much concentration of power. Barclays is now the only leading clearing bank not to have a separate chairman and chief executive.
In view of its recent troubles, Barclays' traditional structure cannot be treated as sacrosanct, and either Sir Peter or Sir Martin would be acceptable to the institutional shareholders since most of the chairmen of the other main clearing banks have no background in banking. Only two - Sir Jeremy Morse at Lloyds and Rodney Galpin at Standard - have any previous banking experience.
Sir Peter and Sir Martin will face opposition, however, from senior officials at the clearing bank.
If the institutions prevent Mr Buxton from taking up the chairmanship, they could end the dominance of the 'Barclays families', which have controlled the top jobs in the bank for generations.
Institutional frustration with Barclays was sharpened by last week's results, which included provisions of pounds 1bn. Sir John admitted that the final dividend might have to be cut.
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