Clooney, who co-starred with Bing Crosby in the film White Christmas, had been battling a recurrence of lung cancer.
She died shortly after 1800 (0100 BST) on Saturday surrounded by members of her family.
Clooney was the aunt of actor George Clooney, former star of TV hospital drama ER, and she made two cameo appearances in the programme alongside her nephew.
She underwent surgery at the Mayo Clinic in Minnesota in January to have the upper lobe of her left lung removed.
"She was a singer who made an incalculable contribution to American popular song by her extraordinary and wise interpretations of these classics," entertainer and composer Michael Feinstein said.
Early role
Clooney's career started in the 1940s and her early hits included duets with Guy Mitchell, Frank Sinatra and Marlene Dietrich.
Born in Maysville, Kentucky, on 23 May 1928, she started singing with her younger sister, Betty, on WLW radio in Cincinnati in 1945, earning a salary of $20 each.
She soared to fame in 1951 with her recording of Come on-a My House, and subsequently became a star in both television and films.
Career comeback
She starred with Bing Crosby in the 1954 film classic White Christmas, and later made an album, Fancy Meeting You Here, with the crooner.
Clooney's hits
Half As Much (1954)
Man (1954)
This Ole House (1954)
Mambo Italiano (1954)
Where Will The Dimple Be (1955)
Hey There (1955)
Mango (1957)
She married Oscar-winning actor Jose Ferrer and her career was side-lined as she entered a period of semi-retirement to raise her five children.
The pair divorced and her attempts to return to performing were damaged by problems with drink and drugs.
But in the 1980s, she staged a dramatic comeback starting a new career in a purer jazz vein, making a series of highly regarded albums for Concord records.
Clooney received a lifetime achievement award at the annual Grammy music awards in February.