Michael Rake, deputy chairman of Barclays Plc
Rake, who was considered favourite for the job, is not interested in the role, three people familiar with the matter said on Sunday.
The next Barclays chairman faces a stiff challenge. The bank was fined $450 million three weeks ago for manipulating Libor interest rates, and the scandal has unearthed deep problems in its relations with regulators, who have accused the bank of frequently being too aggressive.
The job will also attract intense scrutiny -- and possibly interference -- from UK authorities, who have been criticised for not doing more earlier to rein the bank in.
Rake would have been forced to give up his chairmanship roles at telecoms firm BT Group
He could not immediately be reached for comment.
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Marcus Agius, chairman for 5-1/2 years, will step down once a replacement is found.
Barclays is also searching for a chief executive. Bob Diamond, who was CEO from the start of 2011, quit with immediate effect a week after the Libor settlement, following a political and public backlash. When he left, Agius took on the top executive responsibilities.
The fallout from the scandal is expected to overshadow the bank's half-year results on Friday, which should show it made an underlying profit of 3.8 billion pounds, according to the average of analysts polled by the bank.
Rich Ricci, head of Barclays' investment banks, is not interested in being chief executive, a person familiar with him said. He was not seen as a likely replacement for Diamond anyway, as he had been his key lieutenant for many years.
Barclays is expected to look for external candidates to fill both roles.
Headhunter Spencer Stuart is helping Agius in the search for a CEO and Ana Mann's MW is helping John Sunderland, a non-executive director at the bank, run the chairman search.
CEO candidates include former Barclays finance director and a current advisor Naguib Kheraj; Antony Jenkins, head of its retail business; and Bill Winters, former co-head of J.P.Morgan's
Candidates for chairman include former top civil servant Gus O'Donnell and Glen Moreno, the former Lloyds Banking Group
Anthony Sal, a veteran city lawyer, is expected to be picked this week to lead an investigation into the Libor scandal that the bank has pledged to conduct as it seeks to rebuild its reputation.