The house, 25 Gloucester Circus, was designed by Georgian architect Michael Searles in the late 18th Century.
The couple who own the property said the extension was “essential” to modernise the house and to make it their “dream family home”.
However, those against the extension say it would “obscure the original design intent”, and it would mean the last remaining one-storey double entrance would be lost.
The next door neighbour said it would be “very sad” to lose the joint entrance because it was “so unique, the last of its kind”.
“It is a regular stop on guided walks of Greenwich. Groups of people stop outside and it’s lovely,” she said.
“People point at how beautiful the house is, particularly the gap, and they want to take photos of the two front doors.”