U.S. Secretary of State Rex Tillerson says President Donald Trump told him to "begin a reengagement process" aimed at rebuilding relations with Moscow and not to let the turmoil in Washington over possible Russian ties to Trump's inner circle get in the way.

Tillerson made the comments in New Zealand on June 6, amid probes by the FBI and Congress into Russia's alleged meddling in the U.S. election last year and any signs of collusion by Trump's campaign and transition team.

Trump "has been very clear with me that Russia is an important global player, and today our relationships with Russia are at a very low point and they've been deteriorating," Tillerson said.

"So the president asked me to begin a reengagement process with Russia to see if we can first stabilize that relationship, so it does not deteriorate further, and then can we identify areas of mutual interest where perhaps we can begin to rebuild some level of trust and some level of confidence that there are areas where we can work together," he said. "And that's the process that's under way today.

"The president has been clear to me: 'Do not let what's happening over here in the political realm prevent you from the work you need to do in this relationship'," Tillerson said.

He added that Trump had told him "to proceed at whatever pace and in whatever areas we might make progress."

The top U.S. diplomat spoke two days before former FBI chief James Comey is to testify before the Senate Intelligence Committee on June 8 for a public hearing on Russia's alleged meddling in the election.

Tillerson’s comments also came a day after a media outlet published a top-secret document from the U.S. National Security Agency that described repeated efforts by hackers linked to the Russian state to break into U.S. voting systems before the November 8 vote.

The Intercept said the classified document, published online late on June 5, depicts a hacking operation tied closely to Russia's military intelligence agency, known by the acronym GRU, that targeted private U.S. companies providing voter-registration services and equipment to local governments around the country.

The document said that the operation potentially threatened the integrity of the U.S. voting system by trying to trick more than 100 local election officials into opening malicious spyware, and that it went on for months -- ending just days before the election. It did not say whether the operation affected the outcome of the presidential vote.

The Intercept, which focuses on national security issues, said the White House NSA document was dated May 5 and is the most up-to-date and detailed discussion of Russian influence efforts as yet disclosed to the public or press.

After the document was published, the U.S. Justice Department announced it had arrested Reality Leigh Winner, an employee of a national security contractor, for leaking top-secret information dated "on or about May 5" to "an online news outlet."

It said Winner, 25, worked at a facility in the U.S. state of Georgia and was arrested on June 3.

The indictment said Winner was one of only six people to print the document in question and that she had exchanged e-mails with the news outlet, which the indictment did not name but which Reuters reported was the Intercept.

Intercept spokeswoman Vivian Siu said the NSA document was given to the outlet anonymously and "the Intercept has no knowledge of the identity of the source."

The Intercept said that the NSA sought to dissuade it from publishing the document and then requested redactions, some of which were accepted by Intercept.

Russian President Vladimir Putin's spokesman, Dmitry Peskov, denied that Russian military intelligence hackers tried to infiltrate U.S. voting systems.

"Apart from this claim, which absolutely does not conform to reality, we have not seen any other information nor heard any arguments for the reliability of this information and we resolutely deny the possibility that such a thing could have happened," Peskov told journalists on June 6.

The published document expands on allegations laid out by the U.S. intelligence community in January that Putin ordered a concerted effort, involving hacking and disinformation, to tilt the election toward Trump.

It comes days after Putin repeatedly denied any interference in the election in a series of interviews, though at one point he said that there could have been hacking by Russians he said were unconnected to the government.

The NSA report says that "Russian General Staff Main Intelligence Directorate [GRU] actors...executed cyberespionage operations against a named U.S. company in August 2016, evidently to obtain information on elections-related software and hardware solutions."

"The actors likely used data obtained from that operation to...launch a voter registration-themed spear-phishing campaign targeting U.S. local government organizations," it says.

The report shows that, by trying to steal log-in credentials and using spear-fishing e-mails to plant malware, the hackers "obtained and maintained access to elements of multiple U.S. state or local electoral boards."

How successful the operation was, and what kind of data may have been stolen, is not known, the document says.

The document also notes that despite former President Barack Obama's warning to Putin in September to not interfere with the election, the hacking attack on voter systems continued through October.

With reporting by AP, Reuters, and AFP