Police officers might have questioned runaway teenager Megan Stammers and her maths teacher before they disappeared, according to the Daily Mirror.
Officers were also reported to have seized the pair's mobile phones.
A spokesman has declined to comment on a report.
Police across Europe are searching for Megan, 15, and 30-year-old Jeremy Forrest after they eloped to France last Thursday.
It has emerged that concerns were raised to a teacher at Bishop Bell C of E School, in Eastbourne, East Sussex, about the contact between Megan and Forrest following a school trip to Los Angeles in February.
A friend of Megan's said another classmate raised the issue after seeing the pair holding hands together on the return flight, and they would often hug each other, the Daily Mail said.
East Sussex County Council confirmed it and school officials were "addressing and investigating concerns" that were raised about the closeness of Megan and Forrest.
An East Sussex County Council spokesman said: "Because the investigations are ongoing we are not going to comment in detail.
"However, the school and the county council had been addressing and investigating concerns that had been raised, in line with procedure, when this happened.
"That investigation will continue and we will decide what action to take."
The teenager's mother and stepfather have said they were unaware of any relationship, and Sussex Police was only made aware of it last week.
Megan's family has said she had been receiving extra-curricular maths lessons since before the summer, but they did not suspect any relationship between her and Forrest.
Megan's stepfather Martin Stammers, 43, told The Sun that he worked with Forrest two years ago at The Causeway School in Eastbourne, where he teaches.
Mr Stammers told the newspaper: "I knew Forrest as a colleague. He worked at a twin school.
"He did some work for us about two years ago. He seemed a nice enough fellow - there was nothing out of the ordinary about him.
"But this is all about Megan and getting her back - then the police can deal with him."
The criticism came as pictures were released of Megan, from Eastbourne, and Forrest on board a ferry bound for France.
In one of the images issued by Sussex Police, the pair are seen hand in hand on the Dover to Calais route at 9.30pm last Thursday.
The pair's disappearance has sparked a Europe-wide search, prompting frantic appeals by her family to make contact.
Police said that although they entered France, it was possible they were elsewhere as they renewed their plea to the public to come forward with sightings.
Chief Inspector Jason Tingley of Sussex Police said: "We are releasing this photo to show Megan and Jeremy's appearance at the time they travelled to France as we believe it will improve the public's ability to recognise the pair together.
"Although they entered France, it is possible that they are elsewhere now.
"Therefore our objectives are to identify where they are and for them to contact us, so we can bring Megan home safely."
Mr Tingley said they were working with Interpol, a liaison officer in France and the French Missing Persons Bureau, plus other law enforcement agencies in Europe.
He went on: "At this time our foreign liaison is focused on raising awareness of the pair and gathering information about where they could be. We have the ability to request other support and resources from the European authorities when the need arises.
"On an inquiry like this, there will always be reports of unconfirmed sightings, which we will follow up.
"We will not comment on speculation about locations, but with public awareness being a key factor in this inquiry, we will publicise confirmed sightings.
"We would like to thank the public and media for their assistance. We are particularly grateful for the efforts of the international media in raising awareness about the pair abroad. Our primary concern is still to find Megan and Jeremy safe and well."
Officers believe friends might hold clues about where Megan and Forrest might be. Mr Tingley said they should not feel scared of divulging information not previously disclosed.
He told BBC Radio 4's Today programme: "We do know that (Megan) has spoken about perhaps going somewhere but we just need to get really into the specific detail of that."
Megan left her home in Walton Close, Eastbourne, last Thursday afternoon after telling her mother she was going to stay overnight at a friend's house.
Later that day, CCTV caught Forrest's black Ford Fiesta heading to the Port of Dover in Kent with Megan in the passenger seat.
They headed across the English Channel on a ferry bound for Calais some time after 9.30pm.
Megan and Forrest, an amateur musician who lives in Ringmer, near Lewes, were due to return on Sunday evening but they did not board the ferry.
The last contact was on Friday when a message was sent to one of her friends, saying that she was in France. However, the message did not come from her phone.
Megan had drawn up an online "bucket list" - a compilation of goals someone wants to achieve before they die - in which she crossed out "fall in love" and "have someone write a song about me".
The list of 50 dreams revealed her hopes to write a book, fly in a hot air balloon, meet singer Lana Del Rey, go to Glastonbury Festival - and get married.
On her profile, she says she likes to be referred to as "Meg" or "Megzy", and writes about her dream of becoming a musical theatrical actress.
She adds: "I adore music, high fashion, art, photography, blogging, tattoos, make-up, graphic design, travelling, writing, reading and Marilyn Monroe."
As the search for Megan continued on the continent, it was also revealed that two men with links to the Bishop Bell school have been accused of child sex offences.
In 2009, supply teacher Robert Healy, then aged 27, was jailed for seven years at Lewes Crown Court after grooming two pupils on social networking site Bebo.
A former chairman of governors, Canon Gordon Rideout, 73, is due to appear at Crown Court next month charged with 38 child sex offences over an 11-year period from 1962. The school has previously acknowledged the links but said he was not a member of staff and the allegations were historical and unrelated to the school.
Megan is white, 5ft 6in tall, slim, with long, straight, dark brown hair, blue eyes and was last seen wearing a white vest top and silver chain.
Forrest is 6ft, slim with fuzzy facial hair and mousy brown hair. He wears square glasses, dresses casually and has a black star tattoo on his upper left arm.
Anyone with information is asked to call Sussex Police on 101, quoting Operation Oakwood.
A dedicated telephone number and text number which can be used from abroad have been set up. They are 00 44 1273 475432 or text +447786 208090.
Sussex Police said this morning that Megan and Forrest had not been found overnight.