Mueller found no evidence of collusion between Trump's campaign and Russia in the 2016 election, according to a summary of the special counsel's key findings made public on Sunday by Attorney General William Barr.
In a letter to key members of Congress, the top Justice Department official said Mueller wrote that the report "does not conclude that the president committed a crime, it also does not exonerate him."
"So, Mueller's long-awaited report proved what was known in Russia from the very beginning: There was no collusion between Trump and any of his team with the Kremlin," Konstantin Kosachev, a senator in Russia's upper house of Parliament, said in a Facebook post on Monday.
"Two years of unceasing lies. Two years of the highest-level policy built on the allegation of collusion. A conspiracy explaining the allegedly pro-Russian position of Trump, because of which he was essentially forced to impose more and more stringent measures against our country."
"That is why two years were not just lost for Russian-American relations, but simply crushing for them. Someone will answer for this damage? Someone apologize? Someone will adjust something?" Kosachev added.
While the special counsel himself did not determine whether Trump obstructed justice by trying to influence the investigation, Barr said he and Deputy Attorney General Rod Rosenstein concluded there was insufficient evidence to charge the president with obstruction.
Barr also said that the probe did not conclude that Trump or anyone in his campaign "conspired or coordinated" with Russia in its influence campaign.
"Accusations of Russian interference is not news for us, at all levels such narrative has been presented several times and Barr's evaluation of Mueller's report does not make any changes," a Kremlin source, who did not want to be identified, told CNBC.
Trump, who repeatedly called the probe a "witch hunt," claimed victory after Barr's letter became public. But the attorney general's summary managed to spark new controversy, particularly about how Barr arrived at his conclusions in two days following a nearly two-year investigation.
"We saw some sort of squeeze, some summary (of the Mueller report), which, however, does not say anything new — except for the recognition of the absence of some kind of collusion," Kremlin spokesman Dmitry Peskov said at a weekly press conference on Monday, according to a translation.
"It is difficult to find a black cat in a dark room, especially if it is not there," Peskov said, referencing Chinese philosopher Confucius.
Trump may be cleared of obstruction in the opinion of the Justice Department, but other investigations into his conduct will go on.
"From the very start it was a biased, artificial, provocative, conspiratorial, designed-to-fuel-hatred towards Trump campaign. Its second goal was to demonize Russia and prevent any U.S. moves towards better relations with Moscow," Russian senator Alexei Pushkov tweeted on Sunday.
The four-page summary published by Barr on Sunday is set to intensify Democrats' public lobbying to see Mueller's full report over the coming days.
The Democratic-held House is also widely expected to call on Barr to testify on how he came to his conclusion on whether the president obstructed justice.
"In short, Russia did not elect Trump," Evgeny Popov, the co-host of a pro-Kremlin talk show, said via Twitter on Sunday.
"Mueller did not find evidence of collusion between Trump's team with Russia. The U.S. Attorney General submitted a 4-page report on the special prosecutor's investigation report to Congress. D (is for) Disappointment," Popov said.
— CNBC's Jacob Pramuk contributed to this report.