Stiles, whose robust game earned him the nickname the "Toothless Tiger," had suffered from advanced dementia and prostate cancer.
He was one of only four English players to win both the World Cup and European Cup alongside former United teammates Bobby Charlton, Jack Connelly and Liverpool's Ian Callaghan.
A statement from the Stiles family read: "The Stiles family are sad to announce that Nobby Stiles passed away peacefully today surrounded by his family after a long illness.
"The family kindly ask for privacy at this sad time."
Stiles made 28 international appearances for England and was part of the 1966 winning squad.
We are extremely saddened to learn of the passing of Nobby Stiles MBE.— Manchester United (@ManUtd) October 30, 2020
An integral part of our first-ever European Cup-winning side, Nobby was a titan of the club's history, cherished for his heart and personality on and off the pitch.
He will be sorely missed by us all. pic.twitter.com/aT0Mk47cEl
He also won two league titles and the 1968 European Cup during his 11-year spell with United.
The England national team paid tribute to Stiles on social media: "We're incredibly saddened to learn of the passing of Nobby Stiles, a key member of our @FIFAWorldCup winning squad, at the age of 78.
"All of our thoughts are with Nobby's loved ones."
The midfielder made nearly 400 appearances for United from 1960-1971 before he moved to Middlesbrough and later finished his career as a player-coach at Preston North End.
However, he longed to return to Manchester and soon took up a role as a youth coach at United under Alex Ferguson, working with academy players -- such as David Beckham and Ryan Giggs -- who would go on to be known as the famed "Class of '92."
United also released a statement which read: "We are extremely saddened to learn of the passing of Nobby Stiles MBE.
"An integral part of our first-ever European Cup-winning side, Nobby was a titan of the club's history, cherished for his heart and personality on and off the pitch. He will be sorely missed by us all."
Following his retirement, Stiles managed Preston North End and West Bromwich Albion with a three-year stint with Canadian side Vancouver Whitecaps sandwiched in between.
Information from Reuters has been included in this report.