Every last, tenuous lead had been followed to its conclusion, every unlikely theory had been explored. Witnesses were re-examined, morsels of evidence painstakingly re-evaluated. Yet still there was nothing.

So, by the end of this summer, the last rites in Operation Grange — the Metropolitan Police’s five-year, £12 million investigation into the disappearance of Madeleine McCann — were expected to have been played out.

Met chief Sir Bernard Hogan-Howe warned that the end was approaching in a radio interview two months ago. The team of detectives working on the case had been cut from more than 30 to two or three, he said, and when they had followed one remaining line of inquiry, the plug would have to be pulled.

This week, however, there has been yet another astonishing twist — one that has given the dying inquiry fresh impetus.

The late broadcaster, humorist, politician and chef Sir Clement Freud at a party given by the publishing executive Victor Lowndes

Police are investigating what Freud knew about the disappearance of Madeleine McCann

Last Wednesday, I can reveal, two detectives from Operation Grange were quietly dispatched to Somerset to pursue a new avenue of inquiry. They spent 90 minutes interviewing Vicky Haynes, the 64-year-old grandmother who, in a haunting Daily Mail interview last week, told how she had been sexually abused as a 14-year-old girl by the late broadcaster, humorist, politician and chef Sir Clement Freud.

For Mrs Haynes suspects that Freud — who owned a secluded villa in the Algarve resort of Praia da Luz, a few hundred yards from the Ocean Club holiday complex from which Madeleine vanished in May 2007 — harboured crucial information about her fate.

And she fears that when the celebrated former Liberal MP died, aged 84, in 2009, he may have taken this grisly secret to his grave. Indeed, her son, Mason, told me he first emailed the Operation Grange team, to alert them to this disturbing suspicion, some three years ago, after they learned that Freud not only kept a holiday home in Praia da Luz, but that he invited Gerry and Kate McCann to lunch there, just two months after Madeleine went missing.

Mason is nobody’s fool nor a conspiracy theorist. This towering man is one of the world’s most respected close-protection guards.

Yet he says the Grange team either ignored or forgot about his message, and ‘apologetically’ retrieved it only this week after his mother told her story to the Mail.

‘When I heard about the connection between Clement Freud and the McCanns, a light-bulb was switched on in my head,’ he told me. ‘It makes no sense for a man we now know to have been a predatory paedophile to have invited them into his life, at that time, unless he had some ulterior motive.

Madeleine McCann on the day she went missing from the family's holiday apartment in Praia da Luz, Portugal in 2007

‘My theory is that if you live somewhere for 25 years [the length of time Freud owned his Algarve villa] as a paedophile, you become part of the paedophile ring that exists in that area. You share information and stories; that’s how it works.

‘I think the abduction was carried out by that ring, and Freud knew something about what happened to the child. Either he invited Madeleine’s parents to his house to get some sick turn-on out of their visit, or he was trying to get information about where the police had got to with the investigation.

‘It is well-known by psychologists that paedophiles like to go back to the scene of their crimes. Perhaps this was his way of doing it.

‘Freud was 83 at the time. I’m not suggesting he went to that flat and abducted Madeleine himself. But you never stop being a paedophile, no matter how old you are, and I think he knew something.’

With those words in mind, it is chilling to recall that Kate McCann once described a nightmare in which she dreamt Madeleine was buried on a hillside on the outskirts of Praia da Luz . . . close to Clement Freud’s villa.

Many years ago, before the rugged western tip of Portugal morphed into a package-holiday destination favoured by middle-class families such as the McCanns, it attracted a very different type of Briton.

With its donkey-paced ambience, rustic villas that were as ridiculously cheap as the delicious local wine and fish, and a relaxed moral code (described by one long-time expat as ‘anything goes, as long as nobody knows’), it was then a magnet for Bohemian celebrities and minor aristocrats.

Among those lured to this furthermost tip of the Continent was Freud, whose diverse talents — including being the hangdog face of the nation’s favourite pet-food commercial (juxtaposed with his lookalike bloodhound, Henry) — had made him a national treasure.

The father-of-five also wrote children’s books and became a fixture for a time on the much-loved after-school programme Jackanory.

Perhaps not coincidentally, he was a friend of the paedophile entertainer Rolf Harris, too, having given the Australian his entrée into the London club scene when he managed a West End nightspot. And after being elected as the maverick Liberal MP for the Isle of Ely, in 1973, he shared a Commons office with another serial child abuser, Cyril Smith.

Freud had homes in St John’s Wood and Suffolk. In 1987, after losing his parliamentary seat and acquiring a knighthood, he also bought a white-washed villa on the promontory overlooking Praia da Luz bay.

Sir Clement Freud’s villa in Praia da Luz and, right, the paedophile MP relaxes at a party in the Seventies

This cliff was later one of the places where Kate and Gerry McCann went jogging to ease their angst as they waited for news of Madeleine.

Freud often chronicled his summer exploits in Praia da Luz in his weekly column for The Times; and given how he chose to befriend the McCanns, we might find these jottings quite pertinent. While he waxed lyrical about his herb garden and citrus groves, which provided ingredients for his favourite Mediterranean recipes, and we learnt that he played golf and boules, he wrote little about the company he kept.

In fact, the only people he mentioned were his Portuguese ‘Maria’ — the maid who came each day to wash, clean and iron — and his secateurs-wielding gardener, Alfredo.

Was this perhaps because Freud didn’t want anyone to know about the dark company he kept?

After all, as the Madeleine inquiry revealed, the western Algarve, with its bucket-and-spade resorts, wide-open verandas and patios, and laid-back security, had become a teeming hive of paedophile activity.

One investigator said he knew of 38 child sex abusers living in the area; and between 2005 and 2009, the children of seven holidaying families were reportedly subjected to sex attacks — five before three-year-old Madeleine was taken from Apartment 5A at the Ocean Club, and two afterwards.

The names of paedophiles and suspected paedophiles based in Praia da Luz or nearby towns when Freud was a regular visitor are too many to mention. They include serial sex offender Raymond Hewlett, a Briton who was living on the Algarve with his wife and six children in May 2007, and became a prime suspect shortly before dying of throat cancer, aged 64, four years ago.

Another Briton, Chris Ridout, worked as a DJ in his parents’ pub, the Plough and Harrow, 200 yards from the Ocean Club. He was forced to flee the resort shortly before Madeleine disappeared, after he was alleged to be bombarding a 12-year-old local girl with lewd texts and indecent images. He is now thought to be living in the United States.

Ridout doesn’t have any convictions and says he was unfairly hounded after death threats.

Freud is said to have drunk on occasion in the Ridouts’ pub, which locals dubbed ‘The Plough and Paedo’ (daubing its new name in huge letters on the wall). Could it be that the knight of the realm was secretly consorting with loathsome men such as Hewlett?

Freud befriended the McCanns in Praia da Luz, in the western Algarve

Of course, Mrs Haynes and the three other women who have come forward in recent days to describe being assaulted by Freud were far older than Madeleine when he forced himself on them, and paedophiles typically choose their victims from a narrow age bracket.

Yet, as one expert told me, they also tend to gravitate towards others who share their proclivities, regardless of the type of victims they prefer.

Freud retained the five-bedroom villa, Casa da Colina, with its shaded pool, sunken terrace and pastoral tiles, until 2002, when he sold it to the current owner, Andrew Wright.

However, Mr Wright, from Devon, who now rents the property for £2,470 a week in high season, continued to make it available to Freud until he died. Intriguingly, he revealed that he had been contacted by Operation Grange detectives ‘about two years ago’, when he confirmed ownership of the villa.

This could suggest the Met squad hadn’t completely ignored the email sent by Mason Haynes after all.

The case is also being reviewed by a fresh team of Portuguese police officers based in Porto, in the north of the country. Yesterday, a source in the unit said they would look into the possible involvement of Freud — who was not mentioned once in the original 11,000-page case file.

‘This person [Freud] was never under investigation during the case, not even under British formal request,’ said the source. ‘This new information will be considered in the review we are now undertaking.’

And so to Madeleine’s disappearance. What evidence is there to suggest Freud might have been involved, however tangentially?

Thus far, the Freud family’s only comment has come from his son Matthew, a high-powered PR. Between expletives, he told a Mail reporter his father had been in Britain on May 3, 2007, the night she was abducted from her bed as her parents ate a tapas supper with friends in a nearby restaurant.

His assertion appears to be supported by the fact that Freud’s horse-racing column for the Sun was published two days later, on May 5.

By early July he was once again installed in Praia da Luz. As Kate McCann recalls in her book, at the beginning of that month they received a cryptic letter at their lodgings near the Ocean Club.

Rosemary Rimmer-Clay claimed she was attacked by Sir Clement Freud when she appeared on ITV News

‘Dear McCanns,’ it began. ‘I have a house in P da L, been ashamed of the intrusion into your lives by our media . . . and if you would care to come to lunch/dinner at any time before Wednesday next, do ring and let me know.’

Doubtless sure of his culinary renown, he signed off with a droll aside: ‘I cook decent meals.’ Ironically, however, Kate and Gerry could barely remember who he was, and had to be reminded. Even so, they were touched by Freud’s gesture of ‘kindness and friendship’ (it came at a time when the tide of public opinion was turning against them, and they were falling under the unwarranted suspicion of the Portuguese police) and duly accepted his invitation.

Yet this week, when I asked several key players in the McCann saga what they made of Freud’s decision to place himself at the centre of the circus surrounding Madeleine’s disappearance, they said they found it odd, to say the least.

For Freud was an obsessively private man who always kept himself apart from the English ex-pats in the resort, and he is not remembered for his neighbourly gestures.

Residents described him as ‘reclusive’ and ‘aloof’, but most could barely recall seeing him at all. Was this frigid man really so moved by the McCanns’ plight? Or, as Mason Haynes ponders, was his gesture motivated by sinister self-interest?

‘I’d like to know what he had to gain by the meeting,’ he said.

Clement Freud with daughter Emma Freud in 1983 - police are now probing what he knew about Madeleine McCann

The McCanns attended the luncheon accompanied by Gerry’s sister, Trisha Cameron, and her husband Sandy, and their then PR adviser Justine McGuinness, a former Liberal-Democrat parliamentary candidate. Though McGuinness and Freud had a political affinity, I am assured that the pair had never previously met and she did not fix the meeting.

McCann watchers suggest, however, that some part might have been played by Clarence Mitchell, who briefly handled the couple’s PR before McGuinness and would become their mouthpiece after she quit the role.

This is because Mitchell entered into a consultancy contract with Matthew Freud’s company a few months later. From his office in Brighton this week, Mitchell insisted this was a coincidence.

As Kate describes colourfully in her book, the lunch was an odd affair. Freud’s opening gambit was to offer his guests shots of strawberry vodka, though it was only just midday. He then prepared them, as Kate recalled, a ‘bloody marvellous’ watercress salad followed by mushroom risotto — ‘the best we’ve ever tasted’ — and regaled them with his trademark wit.

It might easily be construed as an innocuous occasion — until we remember that Freud took pleasure in cooking an elaborate meal for at least one of his victims (for whom he prepared a gourmet omelette) before thrusting himself on her.

Shortly after the lunch, eaten alfresco on the terrace, Freud returned to Britain but remained in touch with the McCanns by email, as Kate recounts. The Operation Grange team will no doubt wish to examine these exchanges, which might with hindsight prove telling.

He returned to Portugal on August 31, whereupon — again weirdly, we might think — he phoned Gerry McCann. Freud cracked a lame joke which compared what he said was his ‘poor circulation’ to that of the ailing Daily Express.

A few days later, when Kate and Gerry were in turmoil after being made arguidos, or suspects, by the Portuguese police, Freud invited them to his home again; this time at 9.50pm. Greeting them in his nightshirt, he offered them brandy and made another dry quip.

Sir Clement Freud celebrates becoming the new Rector of St Andrews University in 2003

‘So, Kate, which of the devout Catholic, alcoholic, depressed, nymphomaniac parts is correct?’ he inquired, referring to the plethora of newspaper stories speculating on her character and parlous state of mind.

When the couple told Freud how sniffer dogs were supposed to have detected signs that someone had died in their apartment, he replied laconically: ‘So what are they going to do? Put them on the stand? One bark for yes, two for no?’

To Kate, who could have no idea about the kind of man they were consorting with, this was just the tonic she needed: Freud’s ‘way of making everything seem a little less terrible’.

According to Clarence Mitchell, she and her husband were ‘shocked and appalled’ when they learnt the truth about him a few days ago. Yet, on the basis of all they know so far, he says they ‘can see no reason to believe’ he may have had knowledge of Madeleine’s disappearance.