The source said Russia would be expelled from the World Cup "unless the situation improves".
The Russian team are due to play Poland in a World Cup qualifying play-off semi-final on March 24, and would go on to face Sweden or the Czech Republic on March 29 for a place at the finals in Qatar later this year.
FIFA announced on Sunday that Russian teams would be allowed to continue playing under the name of the Football Union of Russia, playing home games on neutral territory and behind closed doors, and with the Russian flag and anthem banned.
Those measures were dismissed as "totally unacceptable" by Polish FA president Cezary Kulesza, who added that Poland would not play their World Cup play-off with Russia, "no matter what the name of the team is."
Sweden and the Czech Republic followed suit, with Swedish FA chief Karl-Erik Nilsson saying he was "displeased" with FIFA's decision.
The draw for the World Cup is due to be held in Doha on April 1 with the tournament taking place in November and December.
The International Olympic Committee on Monday called on sports organisers to ban Russian and Belarusian athletes and officials from international events "in order to protect the integrity of global sports competitions and for the safety of all the participants".
UEFA, which last week stripped the Russian city of Saint Petersburg of this season's Champions League final and moved the game to Paris, must now decide whether to exclude Russia from the women's European Championship.
Russia are due to play the Netherlands, Sweden and Switzerland in their group at the 16-team tournament in England in July.
Spartak Moscow also face being kicked out of the Europa League.
They are scheduled to play RB Leipzig of Germany in the last 16 next month.
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