The letters include drawings and passages of surreal humour, such as: "Do you realise that if you shaved your nose every day you would soo grow a reasonable beard on it?"
The letters have been held by the Spender family for more than 70 years.
They have a combined estimate price of £28,000-42,000.
Oliver Barker, deputy chairman of Sotheby's Europe, said: "While relatively little is known about Freud's teenage life, the emergence of these letters is a sensational moment, providing a glimpse into the workings of a truly artistic mind.
"More than just letters, they are artworks in their own right. Filled with drawings and watercolours, they show the workings of the artist, reflecting his artistic output at the time."
The letters show Freud's humorous side, addressed to 'Spethan', 'Schuster' or 'Step-hanio' and signed off 'Lucelli', 'Lucio Fruit' or 'Lucionus Fruitata'.
Spender, who died in 1995, was known for his poetry and essays which addressed social injustice and the class struggle.
Freud would go on to be regarded as one of the finest British artists of the 20th Century. He died in 2011.
The letters will be offered at Sotheby's Contemporary Art Day Auction on 2 July.
