Miles Frost is said to have collapsed while out jogging at the family’s home in Oxfordshire on Sunday.
Friends said the financial investor had appeared “in great form” in recent weeks and there was no indication of any health problems.
Sir David himself died two years ago aged 74 after suffering a heart attack on the Queen Elizabeth cruise ship.
Lord Chadlington, a friend of the Frost family, told the Daily Mail: “Miles was a wonderful eldest son who was very well loved by all his friends and family. This is a terrible shock.
“I had breakfast with him last Thursday and he was in great form. There was no signs that anything was wrong.”
He added: “The Frost family are extremely close and they will all miss him terribly, as we all will. It is hard to lose a loved one at any age, but 31 really is difficult to comprehend.”
Mr Frost had two younger brothers - Wilfred, 29, a TV presenter on CNBC’s Worldwide Exchange programme, and George, 28. They all went to Eton.
Mr Frost was a financier and managing partner at Frost Brooks, a London-based private equity group.
His company website says that his personal network “includes some of the most respected names in business and media”, many of whom have invested with Frost Brooks.
He was also a board member of Paradine Productions, Sir David’s TV production company; BizEquity, which enables business owners to gain an online value of their company; and parcel delivery firm Parcel Monkey.
Chris Hurley, chief executive of private equity firm LDC, where Mr Frost worked between 2006 and 2010 on technology, media and telecoms investments, said: “Miles was a charming and competent professional who was extremely popular with his colleagues. We’re all deeply saddened by the news. Our thoughts are with his family and friends.”
Last year Mr Frost, his brothers and their mother, Lady Carina Fitzalan-Howard, were joined by 2,000 guests, including the Prince of Wales, for a memorial service at Westminster Abbey in Sir David’s memory.
Sir David was a television star for more than 50 years and was known for his incisive interviews - most famously with disgraced US president Richard Nixon.
• SEE ALSO Obituary: Sir David Frost