This is not the first time that Trump referred to the Kremlin as a "competitor", despite Moscow's reassurances that Russian Federation views the United States as a "partner".
The US leader and his Russian counterpart are set to meet in Helsinki, Finland, on Monday on the final leg of Trump's European tour.
They are accused of using keystroke reading software to spy on the chair of the Democratic National Committee (DNC) and hack into the party's computers.
Russia's foreign ministry described the allegations as a "heap of conspiracy schemes" meant to "damage the atmosphere" before Monday's talks.
The Kremlin said there was no evidence linking the 12 to military intelligence or hacking.
Mueller is investigating whether Trump's campaign colluded with Russian Federation and whether the president has unlawfully sought to obstruct the Russian Federation investigation.
"The stories you heard about the 12 Russians yesterday took place during the Obama Administration, not the Trump Administration", Trump wrote. The conspirators searched one hacked DCCC computer for terms that included "hillary", "cruz" and "trump", and also copied select folders, including one titled "Benghazi Investigations", the indictment says.More news: Russians accused of hacking Democrats in 2016 US election
Speaking at the conference of the National Association of Secretaries of State in Philadelphia, Nielsen said the US intelligence community does "consistently observe malicious cyber activity from various actors against USA election infrastructure", according to prepared remarks released by the department.
"The administration has a pretty good policy towards Russian Federation, just the president doesn't agree with it", said Michael McFaul, a former USA ambassador to Russian Federation under Obama.
"There's no allegation in this indictment that any American citizen committed a crime", Rosenstein added, although the "conspirators corresponded with several Americans during the course of the conspiracy through the internet".
President Trump is scheduled to meet with Putin in Helsinki, Finland, to round out his recent European diplomatic tour that included a North Atlantic Treaty Organisation summit in Brussels and a visit to the United Kingdom.
After the indictments were announced, top Senate Democrat Chuck Schumer called on Trump to cancel his meeting with Putin until Russian Federation takes steps to prove it won't interfere in future elections.
Referring to Mueller's probe, he said: "We do have a - a political problem where - you know in the United States we have this stupidity going on".
President Donald Trump's persistence in pursuing friendly ties with Vladimir Putin has highlighted a growing disconnect within his administration over Russian Federation policy.
In February, Mueller charged 13 Russians and three Russian companies in an elaborate conspiracy to interfere in the election.
"I would call it the rigged witch hunt", Mr. Trump said.