Gary Hoffman, 47, will take on the chief executive’s job at the Newcastle-based bank in October.

He will get a basic salary of £700,000 plus incentives, to be announced later.

He will also collect £1.2million compensation, paid over the next two years, for loss of bonuses at Barclays where he is vice-chairman.

He has worked at Barclays since 1982, and was chairman of UK banking.

Northern Rock’s executive chairman, Ron Sandler, who has run the bank since he was sent in by the Government, will now step back from day-to-day control.

Hoffman said he was “excited by the prospect of... returning Northern Rock to private ownership as a thriving, stable business”.

Present chief Andy Kuipers, who took the role in December after Adam Applegarth’s departure, will leave next month with an estimated payout of £560,000.