Britain's Sky News reported the man was a Briton, identified by the police as Robert Murat, who shares a villa with his mother which is now being searched by forensic detectives.
Portugal's national news agency, Lusa, quoted unnamed police sources as saying the man was one of three people questioned late Monday after detectives and forensic experts searched the villa, near the hotel from where Madeleine McCann was apparently abducted.
Police officials at national headquarters would not confirm the report nor name anyone under investigation.
The British Broadcasting Corporation reports the other people being questioned were believed to be a German woman and a Portuguese man. The BBC also reports related "police activity" in the U.K., but did not elaborate.
Under Portuguese law, opening a formal investigation is a first step toward possibly bringing formal charges. The legal measure is designed to safeguard a citizen's constitutional rights. It grants the person in question access to potential evidence and allows the person to mount a legal defense.
British media said Murat may have requested "suspect" status in order to gain the legal safeguards and access to any potential evidence against him.
Lusa also reported that police took computers and mobile phones from the villa, which is about 100 yards from the Ocean Club holiday village in Praia da Luz where the toddler was apparently taken from her hotel room May 3.
Detectives questioned the three in Portimao, the nearest large town to Praia da Luz, Lusa reported.
The girl vanished after her parents left her, and her brother and sister, both aged 2, alone while they went to a nearby restaurant within their hotel complex. Last week, police called off their ground search.
Her parents are still at the hotel.
British media have said that promised contributions to a reward fund total some 2.6 million pounds, including reported donations from Harry Potter author J.K. Rowling, tycoon Sir Richard Branson and "American Idol" judge Simon Cowell. Also, celebrities including soccer star David Beckham have made public appeals for help in finding Madeleine.