The man was reported dead at the Yarl's Wood centre earlier.
A Home Office spokesman said Bedfordshire Police had been informed and a full inquiry would be carried out by the Prisons and Probation Ombudsman.
Sally Chidzoy, BBC Look East Home Affairs Correspondent, said a source at the centre told her the man died after drinking an energy drink.
She said: "My source says the man who died had been arrested with his wife and child over an alleged breach of visa regulations at an airport. They are believed to have travelled from the Indian subcontinent.
"I have been told he was staying at the Hummingbird wing. In the morning he was having breakfast with his wife, and was laughing and joking.
"He was given an energy drink and shortly after collapsed and died, I am told."
A Home Office spokesman added: "Our thoughts are with the family at this very sad time."
The spokesman could not comment on claims that he died after drinking an energy drink.

A number of flags and banners were damaged overnight at the camp at Twaddell Avenue.
The damage was reported at about 07:35 BST. Two men aged 21 and 22, and a 20-year-old woman were arrested shortly afterwards.
They have been charged with criminal damage and provocative conduct. The woman has also been charged with theft.
All three are due to appear before Belfast Magistrates' Court on Wednesday.
Loyalists have been protesting at Twaddell Avenue since July, after a decision to restrict an Orange Order parade.

Whyte, who won bronze at the 2014 Worlds, was left out of the initial GB squad for this year's event, but a wrist injury to Liam Phillips gave him a reprieve.
The 23-year-old is joined by Rio Olympian Kyle Evans in the men's senior event, with Beth Shriever, 18, leading the women's junior team.
"I was gutted to miss out but now I'm motivated," he told BBC Sport.
Paddy Sharrock will join Evans and Whyte in the men's elite competition at the Rock-Hill World Championships in USA on Saturday.
Blaine Ridge-Davis joins Shriever in the junior women's team, with Kye Whyte the only junior man in the GB line-up.
Whyte was not selected for last year's Rio Olympics, and admits he has struggled to recover from the "devastating" decision.
"I've had a lot of setbacks with selections and races. It leaves you sad and it's a real struggle because you're left wondering why you trained so hard," he said.
"For me this is the first step towards Tokyo [2020 Olympics] and although people are getting faster and more hungry I'm ready to prove I'm up for the challenge."
Evans, 23, reached the quarter-finals in his Olympic debut in Rio and is hoping the experience will boost his prospects of success at the World Championships.
"The Olympics only come around every four years and it's the World Championships that you grow up dreaming of winning as you get to wear the rainbow jersey," he told BBC Sport.
"I feel like I've been in this grey area and just on the knuckle of breaking into the podium positions but what I've experienced is pushing me on to be an even better athlete.
"Hopefully I'll continue to learn and grown and I'll be pushing hard for the top of the podium in the near future."
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But a number of Leicestershire batsmen got starts and then got out as they lost wickets at regular intervals.
Debutant Callum Parkinson took 3-64 as four Leicestershire men were out in the 40s, before Eckersley's patient innings helped them rebuild.
He hit seven fours as Leicestershire ended the first day on 300-7.
A detainee has died at an immigration removal centre in Bedfordshire, the Home Office has confirmed.
[NEXT_CONCEPT]
Three people have been charged by police over a hate crime attack on a loyalist protest camp in north Belfast.
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British BMX rider Tre Whyte aims to "prove" he deserves the chance to compete in the World Championships this weekend.
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Ned Eckersley hit an unbeaten 73 as Leicestershire enjoyed a decent opening day with the bat against Derbyshire at Grace Road.