She was born in Sheffield to a Quaker family. Her father was the banker Frederic Seebohm (created a life peer as The Rt. Hon. The Baron Seebohm in April 1972), while her great-grandfather was the great economic historian, also called Frederic Seebohm. Glendinning grew up near York and went up to Oxford to study modern languages. In the second year of her degree, she married one of her Spanish lecturers, Professor Nigel Glendinning in 1958. They divorced in 1981. Her second husband Terence de Vere White died of Parkinson's disease in 1994 and she remarried in 1996. She had four sons (before she was 28), including Matthew Glendinning, with whom she coauthored the book Sons and Mothers, and the mathematician Paul Glendinning. Another son, Simon Glendinning, lectures in European Philosophy at the London School of Economics having previously taught at the University of Kent at Canterbury.