"He has been imposed an administrative fine of 3,000 rubles [almost $50]," the source said adding that the fan had apologized for his actions.
The incident took place before Tuesday's FIFA World Cup round of 16 clash between Colombia and England, which was played at the arena.
READ MORE: England Knocks Out Colombia in FIFA World Cup Round of 16 in Penalty Shootout
Earlier on Wednesday, the press service of Moscow's Department of Internal Affairs told Sputnik that the 20-year-old man would face administrative penalties for "violating the rules of behavior during the hosting of official sporting events."
The press service of the local government of Moscow's district, where the Spartak Stadium is located, has since told Sputnik that the writing was washed off on Tuesday night.
The 2018 FIFA World Cup kicked off on June 14 and will continue until July 15, with the games being played across 11 Russian cities.