The mystery man, who was discovered in Langho, Lancashire, is not able to tell officers anything about his life or how he came to be found outside the school.
He has dark hair, is aged between 25 and 35, of slim build, is about 5ft 9in and has two tattoos and a pierced ear, police said. He speaks fluent English, but sounds like he may be of Eastern European descent. A passer-by discovered him near the Langho and Billington St Leonards Primary School on 28 December.
Inspector Graham Ashcroft said: "This is a very rare case indeed. We are trying to identify him so that we can get as much information about his background as possible. We want to find out who he is so we can inform his family and friends."
He is being treated for amnesia at the Royal Blackburn Hospital but does not have any other injuries.
"We have checked with the hospital, the mental health teams and the immigration service. His fingerprints have been taken and put on the Police National Computer, which tells us he's not an offender," the inspector added.
"We need to speak to his family to ensure we, or the hospital, can give him the best possible care."