Featuring in a sale titled ‘On the Shoulders of Giants: A Brief History of Big Ideas’, it is estimated at £15,000-25,000.
The letter is addressed to Bill Cleghorn, one of a group of boys with whom Hawking was close friends at St Albans School – the school which the future theoretical physicist, cosmologist, and author attended from the age of ten.
It dates from April 1968, three years after Hawking achieved his doctorate and shortly after he was awarded a post at the Institute of Theoretical Astronomy at Cambridge, founded by the renowned Yorkshire astronomer Fred Hoyle the year before.
Christie’s said the letter is being "offered by the recipient".
In the letter, Hawking, who had been diagnosed with motor neurone disease five years earlier, reports on a trip to America and an uneasy professional detente with Hoyle.
Hawking writes: “Although I wrote my first paper attacking Hoyle's theory of gravity, I have now got a job at his Institute of Theoretical Astronomy. Quite how it will work out I don't know but my present work does not impinge on his so I hope to avoid a collision. Anyway, it means a considerable increase in salary.”
The letter also contains personal references, such as a mention of his son, Robert, who was aged 10 months and who he describes as “very attractive – at least, we think so and other people seem to agree”.
The Christie’s auction is now live online and bidding closes on November 7.
The sale follows the auction of 22 lots from Hawking’s estate at Christie’s in another online sale in November 2018.