Olivant, the investor group, said they would look at taking a minority stake in the bank and introduce a "core-team", which would include Mr Arnold as chief executive.
But according to the group the proposal is not an offer for shares in the troubled lender.
The move would not involve a break-up or sale and the group would not change the brand name.
The group are working on the principle that nothing is wrong at Northern Rock and that the bank just fell foul to a systematic liquidity crunch.
Mr Arnold is believed to have discussed the proposal with the Bank of England, the Financial Services Authority (FSA) and other regulators.
A spokeswoman for Northern Rock said: "There is a formal process and all participants are invited and encouraged to make their bids."