The seats in the Grand Tier are located five boxes away from the royal box in the Grade I listed building.
Agents Harrods Estates said it was the first time such a box had come on the market in almost a decade and there had "already been a bit of interest".
Buying the box will give the owner access to two-thirds of events.
They will also become a member of the Corporation of the Hall of Arts and Science which helps to operate the South Kensington building.
Nicholas Shaw of Harrods Estates said it was "the most expensive box we have ever offered for sale".
The box is on the western side of the auditorium and has approximately 849 years remaining on its lease.
A five-seat box situated on the second tier, whose previous owners included Earl Spencer and the Duke of Devonshire, went on sale in 2011 for £550,000.
A 12-seat private box in the Royal Albert Hall close to where the Queen sits when she attends events has gone on sale for £2.5m.