Banks helped England win the 1966 World Cup and established himself as one of the greatest goalkeepers in history during a 19-year playing career.
Fellow keeper Shilton, who made 125 appearances for England, joined the growing number of people from the world of football to offer their condolences on Twitter.
The 69-year-old wrote: "I'm devastated – today I've lost my hero our condolences to his family rip Gordon."
Banks won 73 international caps and was a key member of the England side that won the nation's only World Cup title to date.
Sir Geoff Hurst, who scored a hat-trick as England beat West Germany 4-2 in the 1966 final at Wembley, said his former team-mate was "one of the very greatest".
Banks played for Leicester, Stoke and Chesterfield domestically, making 510 league appearances in total, and his former clubs have also paid their respects.
Former Manchester United keeper Peter Schmeichel said Banks was one of his heroes and a "true legend in life and football".