Along with his identical twin Sir Frederick, Sir David built a vast business empire which began with hotels and expanded to include shipping, retailing, and, since 2004, ownership of the Telegraph Media Group.
Together with his brother, he lived in a mock-gothic castle on the island of Brecqhou and was one of the Channel Island's richest residents.
The paper said the Barclay brothers had "operated as one" throughout their business career, while steadfastly avoiding personal publicity and media scrutiny.
They had turned to media ownership in 1992 by buying the weekly newspaper The European, which closed in 1998, while they had also owned The Scotsman from 1995-2005.
After first expressing interest in The Daily Telegraph to its Canadian owner Conrad Black in May, 2003, and while a private deal was overruled by a US court in November of that year, the Barclays eventually acquired the paper at auction seven months later for £665 million.
The paper quoted a colleague of the brothers as saying Sir David was distinct from Sir Frederick in that he was 'more attuned to taking a risk, and Frederick was generally willing to have a look but would never bet the farm'.
UK Prime Minister Boris Johnson, who previously worked as a columnist for the title was among those who paid tribute to Sir David.
He said: 'Farewell with respect and admiration to Sir David Barclay who rescued a great newspaper, created many thousands of jobs across the UK and who believed passionately in the independence of this country and what it could achieve.'
Sir David and Sir Frederick were born into a large family in Hammersmith on October 27, 1934, with David the older by 10 minutes.
Their father, also Frederick was a travelling salesman from Kilmarnock who died when the boys were 13.
David and two of his brothers were evacuated several times during the Second World War, the Telegraph said, and ultimately the twins left school aged 14.
Sir David was always a voracious reader, obsessed with newspapers, business, economics and politics, and who had always said he had been educated at the "University of Life", the paper said.
It was reported Sir David died on Sunday.