Mr Bostock is believed to be the leading internal candidate on a list presented to non-executive directors by chairman Sir Philip Hampton at a meeting earlier this month to discuss who might eventually take over from Mr Hester, according to The Daily Telegraph.
But it is thought Mr Hester will remain in place until the process of returning RBS to the private sector begins. The bank, rescued by the Government during the financial crisis, is 81% owned by the State.
RBS declined to comment, but it is understood the meeting was part of what is seen as good corporate governance to ensure a succession plan is in place.
Mr Bostock, 51, will take over as finance director at RBS in October.
Names from outside said to be on the succession planning list include Standard Chartered finance director Richard Meddings, 54, as well as National Australia Bank chief executive Cameron Clyne, 45.
None of the candidates are believed to have been approached and it is not thought that Mr Hester, 52, who has been in the job for four years, is planning to leave imminently.