Since the Libor scandal, Sir David Walker has been appointed as chairman while Antony Jenkins will replace Bob Diamond as chief executive.
It is understood that FSA chairman Lord Turner has written to Sir David to emphasise that the culture of the bank needed to change following the Libor scandal.
He has warned the bank should impose tighter risk management procedures and improve governance. However, Sky News said, it has not demanded a reduction in the size of its investment banking business.

The letter is a standard practice communication sent to major banks and financial services firms when there is a change of leadership.
Sir David will also be quizzed by the Parliamentary Commission on Banking Standards, chaired by Andrew Tyrie MP, next Wednesday. This will be the first public session for the commission and will focus on corporate governance in banking.
Sir David has been called to give evidence as the author of a Government-commissioned report into corporate governance in UK banks and other financial industry entities which he wrote in November 2009.
The commission said he would not be speaking in his capacity of chairman of Barclays.