Reuters reports that Rake's decision was announced Monday by airline easyJet, where he is chairman.
Rake would have been forced to give up his chairmanship roles at both telecoms firm BT Group and easyJet to take the Barclays job.
The Financial Times reports that possible successors to outgoing Chairman Marcus Agius, who wil step down as soon as an appointment is made, include:
Lord Davies, former Chairman if Standard Chartered
Glen Moreno, Chairman of Pearson
Lord O'Donnell, former head of the UK civil service
Hector Sants, outgoing CEO of UK regulator the Financial Services Authority
Bill Winters, former co-CEO of JPMorgan's investment banking unit, is also thought to be in the frame.