The defender, who had been battling dementia, made 407 appearances and lifted two trophies for the Tannadice side over 10 years.
His wife Amanda has been campaigning for dementia and Alzheimer's awareness, and met with health secretary Alex Neil last month.
Alzheimer Scotland joined football clubs and fans in paying tribute to Mr Kopel.
Mr Kopel, who was inducted to the Dundee United Hall of Fame in 2011, started his career with Manchester United as a schoolboy in 1964.
He made 12 appearances for the Old Trafford side, including a European appearance against Anderlecht, before moving to Blackburn Rovers in 1969.
He was signed by United manager Jim McLean on New Year's Day 1972, and became a mainstay at the right-back position in a team which won back to back Scottish League Cup honours in 1979 and 1980, the club's first major trophies.
After leaving Tannadice to take up a coaching position with Arbroath in 1982, he returned to United on several occasions to help out with the reserves.
A spokesman for the club said: "Our thoughts are with all of his family and friends at this very sad time.
"An indication of the affection with which he was held by all Arabs is illustrated by the fact that a supporters' club was named in his honour and by the genuinely warm welcome he received at the many functions and events he attended over the years."
Alzheimer Scotland added: "Deeply saddened to hear of the death of Frank Kopel. Our sincere condolences to his family and friends."
Mr Kopel was diagnosed with a mixture of vascular dementia and Alzheimer's disease in 2009.
His wife Amanda mounted a campaign to raise awareness of the strain on carers looking after people with early-onset dementia, and addressed the Scottish Parliament's public petitions committee.
She has called for free personal care for dementia sufferers under 65, and health secretary Alex Neil pledged to bring her concerns to Parliament after meeting the couple at their Kirriemuir home last month.
Dundee United football legend Frank Kopel has died aged 65.