The development comes after the lender was fined a record £290m by regulators in the United States and Britain over the manipulation of the London interbank lending rate, known as Libor, and its equivalent in Europe (Chicago Options: ^REURUSD - news) - Euribor.
Mark Kleinman, Sky's City editor, said the Barclays board is set to commission an independent inquiry into events at the bank following Mr Agius's resignation.
He said: "Board members are throwing their support behind the idea of a probe of what one shareholder described to me last week as Barclays' 'Wild West culture'".
Kleinman also said Mr Agius was also unlikely to leave his post immediately, and appointing Sir Mike will be complicated by the fact that he would need to relinquish his roles as chairman of BT and EasyJet (Other OTC: EJETF.PK - news) .
Mr Agius' departure is also believed to secure the position of Mr Diamond.
Kleinman said: "That is because the Financial Services Authority would not allow the simultaneous exit of the chairman and chief executive of a major bank like Barclays."
Brian Caplan, editor of The Banker magazine, told Sky News that the Mr Agius' impending resignation "almost smacks of desperation... It does smack of a bit of a panic going on.
"I'm not their PR adviser but I definitely think they would have been better to go along and face the music and accept they are in a very tough situation, explain it as best they can and then see how it went from there."
The news of Mr Agius' reported resignation comes as the head of the government's financial regulator calls for tighter laws to deal with failing bankers.
Professor Molly Scott Cato, author of Green Economics, said: "What we see now is senior executives at Barclays playing some sort of chicken game with the FSA.
"British citizens need to know that the FSA is in control of our banks. They issue a banking licence which is a huge privilege enabling banks to make money and make huge profit, but they do that in return for responsible behaviour.
"Barclays executives have shown that they are not responsible and therefore I believe their banking licence should be revoked."