The man was taken to hospital on Friday after becoming unwell, says the Home Office.
The Home Office told BBC News the man travelled to the UK as part of a small boat crossing on 12 November.
The Independent Office for Police Conduct told the BBC it had received a referral from the Home Office, which it was assessing to determine if further action may be required. Inquest, a charity which helps families of people who have died in state-related deaths, also called for an independent investigation into the man's death, which it called "deeply concerning".
It was designed to hold migrants who had just arrived in the UK on small boats for 24 hours, and has a capacity 1,000 to 1,600 people.
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