More than 400 people, including new witnesses, have been interviewed by Metropolitan Police officers since a fresh investigation was ordered in May 2011, its top detective disclosed.
Kate and Gerry McCann said they were “greatly encouraged” that the “jigsaw” of what happened to their daughter appeared to be closer to completion.
British police have taken statements from, among others, carers, residents and workers who were at the resort in Praia da Luz, Portugal, at the time of Madeleine’s disappearance in May 2007.
In total 442 people have been questioned by officers or provided statements to detectives in an inquiry spanning 30 countries.
Scotland Yard said its investigation had resulted in 4,920 lines of inquiry, of which 2,123 had been followed up and “completed”.
The release of details on the progress of the £5 million police inquiry is part of a coordinated effort by Scotland Yard to maximise publicity ahead of a live television appeal by Mr and Mrs McCann on the BBC’s Crimewatch.
Police sources suggested that they were “optimistic” that they would get to the bottom of what has happened to the missing girl.
Madeleine’s parents said in a statement: “We are greatly encouraged by new information coming to light, with pieces of the jigsaw now fitting together. We are really hopeful that the forthcoming appeal on Crimewatch will bring further new evidence which will take us a step closer to finding Madeleine and to bringing those responsible for her abduction to justice.”
Detectives are studying mobile telephone data from thousands of people who were in Praia da Luz when Madeleine went missing.
Thirty-seven officers are deployed on the case. Police have refused to discuss forensic evidence but admitted that no useful CCTV footage existed in Portugal.
Madeleine vanished from a holiday apartment on the Algarve nine days before her fourth birthday, leading to the biggest hunt in history for a missing child.
Mr and Mrs McCann had left her and her twin siblings in an unlocked apartment while they went for dinner with friends at a nearby bar and restaurant. The McCanns, both doctors who live in Rothley, Leicestershire, believe their child, who would now be 10, was abducted and may still be alive.
Detective Chief Inspector Andy Redwood, who is leading Operation Grange, the new inquiry, said yesterday: “We now believe we have the most complete picture to date of the events surrounding her disappearance.
“We are now making targeted and new appeals for help from the public.
“I truly believe there are people out who hold the key to Madeleine’s disappearance, and that, so far, they may be completely unaware of that fact.
“Through this latest round of publicity we hope to take the next step forward in this investigation. There is a huge public desire out there to help us.”
Following a BBC reconstruction, Mr and Mrs McCann will make a live appeal from Crimewatch’s studio in Cardiff when the programme is aired at 9pm on October 14. Similar programmes will be broadcast in Germany and Holland in an attempt to reach those who were on holiday in Praia da Luz at the time of Madeleine’s disappearance.
There are no plans by Portuguese broadcasters to carry an appeal. Portuguese police and the wider public there have been hostile to the plight of the couple.
When Madeleine vanished there was widespread criticism of local detectives over their conduct. Police in Britain opened up their own official inquiry after a direct plea from the McCanns to David Cameron.
The BBC reconstruction was filmed in Spain rather than in Praia da Luz.
The reconstruction is, according to the Crimewatch team, based on “the Met’s latest, most detailed understanding of what took place around the time of Madeleine’s disappearance.
Robert Mendick, and Fiona Govan in Spain, Telegraph