The tanned, dark-haired man, wearing a distinctive burgundy long sleeve top, is suspected of breaking in to 12 properties where British families were staying in the Algarve, Portugal between 2004 and 2010.
In the four incidents, girls aged between seven and 10 years of age were sexually assaulted.
The attacks happened between 2004 and 2006, before Madeleine vanished in 2007.
Detective Chief Inspector Andy Redwood said tracing the man, said to have "an unhealthy interest in young white female children", is one of his priority lines of inquiry.
His team currently have 38 people classed as "persons of interest" to the inquiry and are also sifting through details of 530 known sex offenders whose whereabouts they cannot account for.
Of those, 59 are classed as high priority and some of those are British.
British detectives are looking at the series of 12 break-ins, in six of which the man sat on or got into bed with young girls, and two of which were in Praia da Luz, where Madeleine was staying with her family when she disappeared.
They had previously been discounted by Portuguese investigators because they are spread over a wide geographical area and there were no apparent attempts at abduction.
Nine of the 12 incidents were reported to Portuguese police at the time they happened, and details of three of those became known to British investigators only after they made televised appeals last autumn.
Mr Redwood said two of the families described the intruder wearing a distinctive burgundy long-sleeved top, and one said that it had a white circle on the back.
In nine of the break-ins nothing was taken, and in the remainder "low-level" items were stolen, such as cash.
The man was described as talking English slowly, with a foreign accent, and remaining calm even when he was disturbed.
"This is an offender who has got a very, very unhealthy interest in young, white, female children who he is attacking whilst they are on holiday in their beds," Mr Redwood said.
"While I completely accept that there are differences (between the break-ins and the McCann case), there is no abduction that we can see, but the assumption from that is that Madeleine McCann has been abducted. That may not necessarily follow with all our thinking about what may have become of Madeleine McCann.
"It is really critical for us to identify this offender and prove or disprove whether he was involved in Madeleine's disappearance."
Mr Redwood said that if names are put forward, his team will be able to eliminate suspects from their inquiry. This suggests that police have DNA on file.
His team had previously appealed for help identifying a man who was seen carrying a child towards the sea on the night that Madeleine, then aged three, vanished, as well as a group of men who were seen lurking near the holiday apartment where her family was staying.
So far they have not been able to eliminate any of the men from their inquiry.