Calvocoressi was born in Calcutta, the only child of Matthew John Calvocoressi (1873–1939). His father was descended from a Greek family from Chios; two uncles married into the Ralli family. He was a director of the Indian branch of Greek merchant business, Ralli Brothers. His mother, Agnes Hermione Melville, was of Anglo-Scottish ancestry and was the granddaughter of Michael Linning Melville. His parents moved to London in 1922. He was educated at Eton College and read Modern Languages at Magdalen College, Oxford.
