The tanned, dark-haired man is suspected of breaking into 12 properties where British families were staying in the Algarve, Portugal, between 2004 and 2010.
In the four incidents, girls aged between seven and ten 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 has 38 people classed as "persons of interest" to the inquiry and is also sifting 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.
The 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.
The man was described as speaking 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," Mr Redwood said.
"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.
British investigators have so far sent three international letters of request to Portuguese authorities over the investigation, covering 41 priority areas for the team, involving 287 separate requests.
Deputy Assistant Commissioner Martin Hewitt said he is frustrated at how slow the legal process has been. "That's causing us frustration because we know what we want to do and we are ready to go with that. But the process is the process."
Another 30 letters have been written to other European countries, but the force would not reveal where.
Madeleine, who was then nearly four, disappeared from her family's holiday apartment in Praia da Luz in the Algarve on May 3 2007 as her parents dined at a nearby restaurant with friends.
British detectives launched a fresh investigation into the youngster's disappearance in July last year - two years into a review of the case - and made renewed appeals on television in the UK, the Netherlands and Germany.
After shelving their inquiry into Madeleine's disappearance in 2008, Portuguese authorities said last October that a review had uncovered enough new information to justify reopening it.