A man who died nine days after being arrested became unwell in the presence of police, investigators say.
Warwickshire Police was called to a property in Nuneaton in the early hours of 10 July by someone concerned about Darren Cumberbatch's behaviour.
He was taken to hospital where he deteriorated and later died.
The Independent Police Complaints Commission (IPCC) said it was reviewing CCTV from the premises and hospital as part of its investigation.
Lawyers for Mr Cumberbatch's family said an incident happened before he was taken to the George Eliot Hospital and it was "possible that a further relevant event" occurred after the 32-year-old arrived there.
The IPCC said: "We know that he became unwell while the police were on the premises and he was then transported to a local hospital.
"We are independently investigating the circumstances of Mr Cumberbatch's death and the contact the police had with him."
In a statement, his family said they were "depending" on the IPCC to conduct a "robust investigation".
"We ask anyone who can give any information about Darren's contact with the police on 9 and 10 July 2017 to contact the IPCC urgently.
"In the meantime we hope the media will avoid speculation as to what happened to Darren and let the investigation take its course."
Warwickshire Police said it had referred the death to the police watchdog and was providing full support to investigators.
Guido Liguor, from the IPCC, said speculation, particularly on social media, about how Mr Cumberbatch died and police interaction with him was "unhelpful".
"I would like to reassure all of those concerned, that we will thoroughly and independently examine the circumstances surrounding his tragic death in order to establish the facts," he said.