In the letter, which was sent to Airbnb, Visa and Samsung, among others, the government emphasizes the close connection between politics and sports and that "Russia and Belarus should not be allowed to use sports as a propaganda tool".
In January, the IOC opened the door to allowing Russian and Belarusian athletes to compete in next year's Olympics in Paris.
This week it was also announced that the International Fencing Federation has voted to allow participants from Russia and Belarus back into international competition.
Britain was one of 35 countries to sign a protest last month against the International Olympic Committee's (IOC) decision ahead of next year's Olympics.