The scheme would entail an arched passageway driven through tenements to link the Royal Mile with a public square. It follows widespread opposition to Mountgrange's initial plan to demolish some of the flats.
"Our preferred option is to remove the tenements but we can accept the alternative application. We are happy to let the council decide between them," said Mountgrange chief executive Manish Chande.
The scheme includes a hotel, a conference centre and residential, retail and cultural quarters. Both proposals will be put before Edinburgh City Council.