The duke, whose portfolio included the five-red-AA-star Chester Grosvenor hotel - managed by Bespoke Hotels -suddenly became ill on his Abbeystead estate, a spokesperson confirmed.
Grosvenor Estate also owns the freehold of Chris Corbin and Jeremy King's Beaumont hotel in London, as part of its widespread property portfolio in Mayfair and Belgravia.
In addition, the duke owned real estate in Belgravia and Mayfair, two of London's most expensive areas.
A spokeswoman for the Grosvenor Estate said: "It is with the greatest sadness that we can confirm that the Duke of Westminster, Gerald Cavendish Grosvenor (64) died this afternoon (Tuesday) at Royal Preston hospital.
"He was taken there from the Abbeystead estate in Lancashire where he had suddenly been taken ill. His family are all aware and they ask for privacy and understanding at this very difficult time.
"No further comment will be made for the time being but further information will follow in due course."
The Chester Grosvenor posted the following statement on its website: "We are all deeply saddened to learn of the death of the Duke of Westminster. Bespoke hotels have only just become part of the Grosvenor story but already feel such a sense of belonging; due to the way in which the Duke fostered care and warmth amongst all his staff and associates. The entire team at the Chester Grosvenor and across Bespoke hotels join us in regret, reflection and respect for the life of a remarkable man."
Bespoke Hotels took over the management of the Chester Grosvenor in April 2015 following the retirement of former managing director Jonathan Slater.
Grosvenor Estate dates back to the 17th century, when the family invested heavily in London property after Sir Thomas Grosvenor married the heiress Mary Davies in 1677.
The Duke of Westminster was one of the richest men in Britain, with his fortune being estimated at $10.8bn (£8.3bn) by Forbes magazine.
In this year's Sunday Times rich list his wealth was reported as £9.35b, putting him in sixth place after an increase in his fortune, although it is believed around £769m was wiped off his fortune after the UK voted to leave the EU on 23 June.
His company also has substantial holdings in Scotland and continental Europe.
Buckingham Palace said: "Her Majesty the Queen is aware of the news about the Duke Westminster. A private message of condolence is being sent from the Queen and the Duke of Edinburgh to the family."
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