The banking giant has appointed headhunters to draw up a list of candidates for the role, although Sir David is not set to leave until next summer after serving a three-year term, according to The Sunday Times.
Sir David – a former executive director of the Bank of England – joined the Barclays board in September 2012 and officially took over at the helm two months later amid a reputational crisis in the wake of its £290m Libor rates rigging settlement.
He replaced Marcus Agius, who resigned following the scandal, which also claimed the scalp of ex-chief executive Bob Diamond.
The City veteran’s successor will face ongoing challenges at the bank as it comes under renewed fire over pay and the impact of a wide-ranging costs and culture overhaul.
Barclays fuelled controversy over banker pay last week when it announced a 10 per cent hike in its staff bonus pool to £2.4bn despite reporting a 32 per cent drop in underlying annual profits.
The bumper payout will see its 26,200 investment banking employees share out a £1.6bn bonus pot for 2013, up 13 per cent on 2012, giving an average payout of £60,100 per employee in the division.
Barclays declined to comment on the reports.