Herd played a key role in United’s revival following the 1958 Munich air disaster and scored twice in their 3-1 win over Leicester in the 1963 FA Cup final.
He also helped United win the English league title in 1965 and 1967.
Herd, who previously played for Arsenal, scored 145 goals for United.
He left United for Stoke a month after they won the European Cup against Benfica in 1968, a success in which he played little part having suffered a broken leg the year before.
United and Stoke were involved in a Premier League match on Sunday, with players from both clubs wearing black armbands as a mark of respect to Herd.
“Everybody at Manchester United is saddened to hear of the passing of David Herd,” the club posted on manutd.com.
“We can confirm that both United and Stoke will wear black armbands this afternoon to pay tribute to David Herd, who played for both clubs.”
Herd’s former United team-mate Paddy Crerand tweeted: “Sad news about my old team mate David Herd. Great player made my ordinary passes into him great ones. Tremendous shot and strength.”