The incidents led the FBI to send a "flash alert" to election officials earlier this month, asking them to watch for similar cyber-attacks.
The FBI alert, first reported by Yahoo News, did not mention Russia.
However, the authorities have attributed the attacks to Russian spy agencies, NBC News quoted US intelligence officials as saying.
"This is the closest we've come to tying a recent hack to the Russian government," one unidentified official said, adding "there is serious concern" Moscow may be seeking to create uncertainty in the election process.
Although the alert does not identify targeted states, Yahoo News quoted officials as saying they were Illinois and Arizona.
Illinois officials said last month that they shut down their state's voter registration after a hack.
On Monday state officials said the hackers stole data from as many as 200,000 voter records, although they told the Chicago Tribune no such record had been deleted or altered.
In Arizona hackers unsuccessfully tried to breach the voter registration system using malicious software, reports said.
The state shut down the system for nine days beginning in late June, after the discovery of malware on an election official's computer. However, officials concluded the system was not successfully compromised.
Although the Department of Homeland Security was not aware of specific cyber threats against election-related networks, Johnson urged officials to secure their systems, according to a transcript of the call released by the department.
Hackers apparently connected to Russia staged a series of cyber-attacks against US media outlets including the New York Times, reports said earlier this month.
And US officials say Russian intelligence agencies were behind recent hacks into Democratic Party organizations, including the campaign of its White House candidate Hillary Clinton.
Another hack of Democratic National Committee servers resulted in an embarrassing leak of emails last month revealing how party leaders tried to undermine Clinton's Democratic rival Bernie Sanders, leading to the committee chief's resignation.
Experts said they believed two Russian intelligence agencies to be responsible.
Last week Senate minority leader Harry Reid asked the FBI to investigate evidence suggesting Russia may be trying to manipulate voting results in November, expressing concern about a "direct connection" between Republican candidate Donald Trump's campaign and the Russian government.
"The prospect of a hostile government actively seeking to undermine our free and fair elections," he wrote, "represents one of the gravest threats to our democracy since the Cold War."
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