Police believe the man, now in jail for a sex crime, was in the area where the girl, then aged three, was last seen.
Madeleine's parents Gerry and Kate McCann thanked the police, adding: "All we have ever wanted is to find her."
Police are appealing for information about two vehicles owned by the man.
The day after Madeleine vanished, the suspect transferred a Jaguar car to someone else's name.
Madeleine went missing from an apartment on a Portuguese holiday resort on the evening of 3 May 2007, while her parents were with friends at a nearby tapas bar.
Her disappearance sparked a huge and costly police hunt across much of Europe - the most recent Metropolitan Police investigation, which began in 2011, has cost more than £11m ($NZ21.5m).
"Someone out there knows a lot more than they're letting on," said Detective Chief Inspector Mark Cranwell, who is leading the Met inquiry.
The force said it remained a "missing persons" investigation because it does not have "definitive evidence" as to whether Madeleine is alive or not.
However, German investigators have classed it as a murder inquiry.
The London police force said the German authorities had taken the lead at this stage of the case because the German suspect was in custody in their country.
German police told the country's ZDF TV channel the suspect, who is not being named, is a sex offender currently in prison for a sex crime.
The man has two previous convictions for "sexual contact with girls", according to Christian Hoppe from Germany's federal criminal police office.
An appeal on German television was broadcast this evening at 19:15 BST.
Cranwell said the prisoner, then aged 30, frequented the Algarve between 1995 and 2007, staying for "days upon end" in his camper van and living a "transient lifestyle".
He was in the Praia de Luz area where the McCann family was staying when she disappeared and received a phone call at 7.32pm, which ended at 8.02pm.
Police have released details of the suspect's phone number and the number which dialled him, saying any information about them could be "critical" to the inquiry.
The suspect is believed to have been using a Portuguese mobile phone on the day Madeleine went missing.
The phone received a call in the area of Praia da Luz from a second mobile number; from someone not in the area. They want the person who made this call to come forward.
"They're a key witness and we urge them to get in touch," Cranwell said.
"Some people will know the man we're describing today... you may be aware of some of the things he's done," he said.
"He may have confided in you about the disappearance of Madeleine.
"More than 13 years have passed and your loyalties may have changed," he added.
"Now is the time to come forward."
In a statement, the McCanns welcomed the appeal: "We would like to thank the police forces involved for their continued efforts in the search for Madeleine.
"All we have ever wanted is to find her, uncover the truth and bring those responsible to justice.
"We will never give up hope of finding Madeleine alive, but whatever the outcome may be, we need to know as we need to find peace."
Police said the suspect was one of 600 people that detectives on the inquiry, known as Operation Grange, originally looked at, though he had not been a suspect.
After an appeal in 2017, "significant" fresh information about him was provided.
Since then, Met detectives have carried out "extensive inquiries" in Portugal and Germany in order to gather more details about him.
The force said it was trying to "prove or disprove" his involvement in the case and retained an "open mind".