Britain's Press Association says Cooper, who was knighted in 2000, died at his son's house in Surrey two days before his 77th birthday.
Cooper had a 40-14-1 record in a professional career spanning more than 16 years but is best known for flooring Ali, then named Cassius Clay, in a fight in 1963 at Wembley Stadium.
Cooper threw a trademark left hook toward the end of the fourth round, catching the much larger Ali flush on the jaw and sending him through the ropes and on to the canvas.
Ali ultimately won the bout and another fight against Cooper for the heavyweight belt in 1966 at London's Highbury Stadium, both by technical knockouts while trailing on the scorecards.
"Ali is possibly the greatest athlete there's ever been and Henry put up a great performance and just wasn't quite good enough on both occasions - but he's not the only one who wasn't good enough to beat Ali," said Robert Smith, the general-secretary of the British Boxing Board of Control. "For such a small man, he put up some great performances in a world-class context.
"It's not just the boxing and your ability, it's the personality as well. He won the BBC Sports Personality of the Year on two occasions, which is a tremendous feat for a boxer. Everyone called him 'Our Enry,' and he was much loved, he served boxing wonderfully."
Cooper retired in 1971 shortly after losing his British, European and Commonwealth belts to Joe Bugner by a quarter of a point.
"One of Britain's greatest sportsmen Sir Henry Cooper passed away today," WBA heavyweight champion David Haye tweeted. "A true warrior and great human being. Rest in Peace."
Former undisputed heavyweight champion Lennox Lewis tweeted: "R.I.P. Sir Henry Cooper. Former British, Commonwealth and European Champion. My deepest condolences to the Cooper family."
