"Have they started blaming Russia yet for the Washington staff changes?" foreign ministry spokeswoman Maria Zakharova said in a text message sent to AFP.
She did not immediately make any additional comments.
Tillerson on Tuesday agreed that Russia "was likely responsible for the nerve agent attack" on former double agent Sergei Skripal and his daughter in England, speaking on a flight back from Africa.
"We agree that those responsible - both those who committed the crime and those who ordered it - must face appropriately serious consequences," he added.
The US has accused Russia of actively interfering in the 2016 presidential election, stealing Democratic party communications and pushing out disinformation through social media, which Moscow denies.
Despite Donald Trump's initial promises of improved ties, relations between the countries have deteriorated sharply under his administration.
Putin once again rejected accusations of election meddling in an interview with NBC released Friday.
"Why have you decided the Russian authorities, myself included, gave anybody permission to do this?" Putin asked in the interview.
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