Barclays should make an external appointment for its next CEO following the departure of Mr. Diamond, who resigned following an interest rate-fixing scandal last month, the newspaper said, citing the same unnamed source.
Shareholders want to see a "leaner and meaner" Barclays which is less reliant on underperforming parts of its investment bank business, such as equities, the report said.
Barclays declined to comment on the story. RBS couldn't immediately be reached for comment.
Sir David Walker, the newly installed Barclays chairman, told the Sunday Telegraph that successfully filling the CEO post was his top priority and that he would be willing to look overseas for the right candidate.