Christopher Mapleston, 29, is fighting for his life in a critical condition in hospital.
The London tax consultant was out with friends on November 19, but ended up in intensive care with a fractured skull, a bloody gash on his head, and no phone.
DC Terry Martin said: “We cannot be certain whether Christopher”s injury was the result of a fall or an attack.”
The KPMG tax auditor, born in New Jersey, went to Nottingham University and then the School of Oriental and African Studies.
Lewisham police would like to speak to anyone who saw Mr Mapleston, wearing a camouflage jacket and black trousers, between Peckham Rye train station and Arbuthnot Road.