Bannister's family said that he died on Saturday in Oxford, the English city where the runner cracked the feat many had thought humanly impossible.
Bannister's family confirmed his death, the BBC said on Sunday.
Bannister, who was a medical student at the time, clocked a time of 3 minutes 59.4 second at a sports ground in Oxford on May 6, 1954, to beat a record by Swedish runner Gundar Haegg of 4:01.4 that had stood for nine years.
Bannister went on to work in neurology after beating the mile record time in 1954. I felt at that moment that it was my chance to do one thing supremely well.
Coe, who had learned the news just before the launch of a new IAAF Heritage initiative created to showcase the sport's former greats such as Bannister, said:"This is a day of intense sadness both for our nation and for all of us in athletics". It was the first time two men had run under 4 minutes in the same race.
Bannister, who was knighted in 1975, went on to a career as a neurologist and watched through the years as other runners ran ever-swifter miles.
"He was an inspiration to those like me who sought to combine university with worldwide sport".
Additionally, the legendary runner also competed in the 1952 Olympics.
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London Marathon: "For some the word legend doesn't quite cut it. Rest in peace Sir Roger Bannister".
May 6, 1954, Britain's Roger Bannister hits the tape to break the four-minute mile in Oxford, England.
Bannister stopped running to pursue a medical career.
"It's awesome that more people have climbed Mount Everest than have broken the 4-minute mile", Bannister said in an interview with The Associated Press in 2012.
Four-time Olympic champion Sir Mo Farah described Bannister as "always humble, supportive and encouraging" and "an inspiration to so many". He leaves behind his wife Moyra Jacobsson, their two sons, two daughters, and 14 grandchildren.
"At the age of 88 he was still an active supporter of the University and we will miss him enormously".
"There is not a single athlete of my generation who was not inspired by Roger and his achievements both on and off the track", tweeted Coe, who won two Olympic gold medals in the 1980s.
Bannister returned to Oxford in 1985 to become Master of Pembroke College, a position he held until 1993.