Pompeo also said the State Department had written recently to the governing boards of US universities alerting them to threats posed by the Chinese Communist Party.
President Donald Trump has repeatedly claimed without evidence that a rise in mail-in voting will lead to a rise in fraud, though he has expressed more openness to absentee voting.
Russian media reported that Russian citizens also received similar messages on Thursday.
The sources, who spoke on condition of anonymity because a law enforcement investigation is underway, said the hackers accessed the account multiple times between July 12 and Oct. 21 last year.
It said there were open source indications that Russia sought to influence the Brexit campaign but that the British government had not sought deep evidence of meddling.
Democratic presidential candidate Joe Biden on Monday said he would "treat foreign interference in our election as an adversarial act" if elected to the White House.
In his call with Lavrov, Pompeo also raised the issue of election security, the department said in a statement, as the United States readies for nationwide voting on Nov. 3.
The justices took up the administration's appeal of a March ruling by the US Court of Appeals for the District of Columbia Circuit that material must be given to lawmakers.
US Trade Representative Robert Lighthizer said in Senate testimony that Bolton's account was "absolutely untrue."