Furse immediately fell head-over-heels in love and the two quickly became friends.
In 1938, they saw more of each other when Furse was allowed backstage to draw the dancers of the Sadler's Wells Ballet - which would later become the Royal Ballet - including Fonteyn, who was already being hailed as the natural successor to the prima ballerina Alicia Markova.
And they evidently stayed in touch even after the outbreak of war a year later, when Furse joined the Rifle Brigade to work in air reconnaissance and liaison. In fact, she poured out her heart to him.
Although little has been known of their friendship until now, in a sequence of nine letters just acquired by the Royal Opera House Collections, Margot Fonteyn writes to Furse conveying her private hopes and fears for herself and for her art. They provide a startling insight into the internal struggles of one of Britain's greatest dancers.
"All my inspiration to fight myself and become an artist comes from you, because you think that I am so much greater than I shall probably ever be. I am still only half an artist and fighting what is often a losing battle with the other half," she wrote in January 1940.
"I seem to have struggled so much with myself, both about my dancing and my own life, that I know myself as I am and not as I want to be and I am trying to live my life as a grown person, which I must become now and not a dreamer as I was in Paris, full of wonderful thoughts and ideas about myself in the future," she confided a year later.
It is clear from the correspondence, which dates from 1940 to 1942, that Furse was very much in love with the enchanting young dancer.
In return, she wants to be kind and obviously values his friendship as part of a social set with which she had several connections. Patrick was the brother of the actor Jill Furse, who was at one point romantically involved with Rex Whistler, who had designed the Sadler's Wells ballet The Rake's Progress.
The Furses' friends included Cecil Beaton, the society photographer, Stephen Tennant, the aristocrat, and Constant Lambert, the composer and conductor who was the music director for the Sadler's Wells Ballet.
But she never reciprocates Furse's passion and it becomes increasingly obvious from the letters that her reserve stems from her love for someone else.
To begin with, she hints, cryptically, of her need for a mate. "Dancing will never quite complete my life. It is the major thing, though, and I don't imagine that it will ever come second to my love for a person," she wrote in 1940.
Eventually in 1942 she confessed that the object of her affection was Constant Lambert, even though he was married with a child. "I felt that I didn't want to tell you about Constant at all until I had time to talk it all over with him and know what was going to happen," she said.
She and Lambert did embark on a relationship that lasted several years in the 1940s but remained a close secret until her death in 1991. In 1955, she married Roberto Arias, a Panamanian diplomat.
As a glimpse into the personal life of one of the 20th century's greatest ballerinas, the newly acquired correspondence is intriguing. But the letters also shed light on the bigger picture of British ballet during the Second World War, a history soon to be told in an exhibition Dancing Through The War: The Royal Ballet 1939-1946, which will open at the Churchill Museum and Cabinet War Rooms in London later this month.
When war was declared, what was to become the Royal Ballet was only a decade old. The company owes its existence to Dame Ninette de Valois, the dancer and choreographer who assembled a school and a company, which was originally known as the Vic-Wells Ballet, as it appeared at the Old Vic near Waterloo and at Sadler's Wells in north London. From 1931, it was called the Sadler's Wells Ballet after taking up residence at the Wells. But during the war, the theatre was taken over as a soup kitchen and accommodation for East End families bombed out of their homes and, while the school continued in situ, the ballet company moved out.
In a frantic schedule designed to boost the morale of the nation, they performed 10 weeks in London's West End, alternated with 10-week tours across the country, to theatres, garrisons and even factories, where the ballerinas took to wearing more make-up and jewellery to overcome initial resistance.
In the early years of the war, they also toured to France, Belgium and even the Netherlands, from where the company, including Fonteyn, had to escape by cargo boat after the Germans invaded.
When the male dancers and even key figures such as Frederick Ashton, the choreographer, were called up for service, boys from the company's ballet school and foreigners filled their places. De Valois was fiercely nationalistic and would not dream of fighting for her male dancers, even though the playwright George Bernard Shaw argued that artists should be exempted from service to satisfy a broader national good. (The dancer Robert Helpmann, with whom Fonteyn struck up a strong working partnership, was, by luck or necessity, an Australian.)
And, of course, those left behind were forced to dance on the same rations as the rest of the country, which left them starving on occasion. In a letter in 1940, Fonteyn wrote: "Sometimes I am too exhausted to do very well..."
Cristina Franchi, the Royal Opera House's exhibitions manager, says this was not surprising. "She was very young when war broke out - 20 - and she found herself leading this company which performed everywhere, eight performances a week and sometimes three a day in London. It was an extraordinary schedule. They had two weeks of holiday - the rest of the time they were rehearsing and dancing. But what these letters show is how Fonteyn felt about herself as a performer and her impetus was always to give her best."
Judging herself harshly, Fonteyn wrote: "Each performance which is not as good as I can possibly do is a waste of time completely and there are all too many of them."
But one upside of this hectic schedule was the massive fan base that developed, not only among British audiences but with American servicemen who packed London theatres during leave. It was said that the Royal Ballet's success when it began tours of the States was built on these wartime performances for the troops.
During the war, the Royal Opera House at Covent Garden was leased to Mecca as a dance hall. When it eventually reopened in 1946, the lobbying of ballet supporters including the economist John Maynard Keynes (whose wife was a ballerina) combined to secure the Sadler's Wells Ballet a new home in the central London theatre as the ballet company of the nation.
Monica Mason, director of the Royal Ballet, says the war years were extremely important in the company's development.
"In spite of losing many male members of the company to the forces the dancers that remained performed in theatres, barracks and any other suitable spaces the length and breadth of England. During those years, often in the harshest of conditions, many new works were created and a substantial ballet-loving audience was built.
"When, in 1946 after the war was over, the company was invited to reopen the Royal Opera House, it had firmly established itself as an institution worthy of national pride."
Dancing Through The War opens at the Churchill Museum and Cabinet War Rooms on 16 February and runs until 20 May. Admission £11/£9, under-16s free
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