Those behind the scheme said the new gallery, next to an existing duck pond, would give them the chance to reunite the collection with the estate.
The paintings would be mounted on sliding racks which could be stacked away when the venue was used as a performance space, giving flexibility to how works were displayed, according to the application.
The building, with two further gallery spaces at either end, would include features such as high insulation and heat pumps.
Comments on the planning application can be made until 19 September.
The paintings have been kept Southside House, Wimbledon, in south-west London.
Hellens Manor, dating from the Middle Ages, is described as one of the oldest dwellings in England and is a family home.
It hosts Hellensmusic, an annual classical music festival, in May as well as a programme of concerts and music teaching throughout the year which attracts international figures in classical music.