Portuguese police sources say Christian Brueckner refused to respond to questions put to him on their behalf by German officers.
Madeleine McCann's parents said they "welcome the news" Portuguese police have identified Christian Brueckner, a German paedophile, as a formal suspect in her disappearance.
Kate and Gerry McCann said they had not given up hope they would one day be reunited with their daughter, even though the chance of that was "slim".
They spoke out after Brueckner refused to answer questions in his first interrogation. He was repeatedly asked where he was on the night the 3-year-old girl went missing while on a family holiday in the Algarve 15 years ago.
The 44-year-old rapist, who is currently in a German prison, was handed a legal document stating he was an "arguido", or suspect, in the child's disappearance in Praia da Luz.
However, Portuguese police sources say Brueckner refused to respond to questions put to him on their behalf by German officers.
Mr and Mrs McCann said in their statement: "We welcome the news that the Portuguese authorities have declared a German man an 'arguido' in relation to the disappearance of our beloved daughter Madeleine.
"This reflects progress in the investigation, being conducted by the Portuguese, German and British authorities. We are kept informed of developments by the Metropolitan Police.
"It is important to note the 'arguido' has not yet been charged with any specific crime related to Madeleine's disappearance.
"Even though the possibility may be slim, we have not given up hope that Madeleine is still alive and we will be reunited with her."
Brueckner was asked to account for his movements on the night Madeleine disappeared in May 2007. It is understood his lawyer was with him during the questioning.
One of the legal grounds for making him a suspect is a claim that he confessed to a friend he had snatched Madeleine. Mobile phone records are also thought to place him in the resort the night she vanished from her holiday apartment as her parents ate tapas nearby.
Portuguese prosecutors were facing a tight deadline because the country's statute of limitation rules prevent someone from being prosecuted for certain crimes 15 years after they were committed.
Because the sentence for abduction and murder carries a sentence greater than 10 years, Brueckner could not be prosecuted after May 3, the 15th anniversary since Madeleine vanished.
Now he has become an "arguido", there is no time limit to how long he is considered a suspect. The Portuguese authorities insisted the decision to say they had a formal suspect was not connected to charging rules.
He was identified as the possible murderer by German prosecutors in June 2020 but has not been charged.
Jim Gamble, the former head of the Child Exploitation and Online Protection Centre, who worked as part of the UK side of the investigation into the child's disappearance, said he believed charges would now follow.
Describing his "gut feeling" Brueckner was the "best fit", he told BBC Breakfast: "You have proximity, you have opportunity, and you have a profile with regards to an offender that absolutely fits in a way that no others have.
"This is all positive. And from my own position, I wouldn't be surprised if charges did follow."
Sources close to the case said they expected no new developments until results were obtained from ongoing forensic analysis of samples taken from a Volkswagen van Brueckner was known to drive and sleep in while a drifter in the Algarve.
The van had been at the centre of police appeals for information in connection with the case.
One Portuguese source described results from samples as "potentially crucial".
He added: "They could also be the final throw of the dice. At the moment it's a bit of a wait-and-see situation."
Brueckner was told of his suspect status in Oldenburg Prison in northern Germany, where he is serving seven years after being convicted in 2019 for raping an American pensioner in Praia da Luz in 2005.
Rogerio Alves, Kate and Gerry McCann's lawyer, warned in July 2020 that Portugal's 15-year limit on prosecutions meant there were less than two years left to take action against Brueckner.
A Lisbon-based lawyer, who asked not to be named, said prosecutors would have been "acutely aware" of the time limits hanging over the case.
"Our statute of limitations brings with it the probability that within a matter of weeks, the person responsible for her disappearance may never be brought to justice in the country where she vanished even with an arrest and confession."
Madeleine's parents were made "arguidos" in September 2007. But their status was lifted in July 2008 when Portugal archived its first investigation.
The Metropolitan Police continue to treat Madeleine's disappearance as a missing persons inquiry.
What is the evidence against Christian Brueckner?
In June 2020, German prosecutors announced they were convinced Christian Brueckner, a prolific sex attacker, was the most likely culprit in the disappearance of Madeleine McCann.
He allegedly confessed to the abduction while watching a television news report marking the 10th anniversary of her disappearance in 2017.
He is said to have boasted to a friend in a German bar that he had "snatched her" while living in the Algarve and "knew all about" what happened. It is said he then showed his friend a video on his phone of him raping a woman.
He had arrived in Portugal in 1995 as a teenage backpacker who worked in catering in the country's southern resorts.
Around the time of Madeleine's abduction, he was living in a VW campervan and carrying out burglaries, stealing petrol and selling drugs in the area.
After a series of appeals about the van, German police now hope forensic analysis of that vehicle could yield evidence of a link to Madeleine.
Two years ago, a witness claimed the girl was seen in it in Spain around the time of her abduction.
Brueckner's mobile phone placed him in the Praia da Luz resort within minutes of her disappearance on the evening of May 3, 2007. Shortly afterwards he returned to Germany.
His penchant for depraved sex crimes, particularly against children, began in Germany in 1994 when he was jailed as a youth for sexually "abusing a child".
Twenty-two years later, he was jailed in his native country again for sexually assaulting a child while making pornography.
He was convicted in 2019 for his most violent attack when he brutally raped a 72-year-old woman in Praia da Luz in 2005. His victim told the court he appeared to enjoy "torturing" her.