Mr Trump has come under fire in recent days after reports alleged that he had been warned about the bounty system but did nothing in response.
The New York Times reported earlier this week that US spies and military members had warned the US government that Russians were allegedly paying bounties to fighters who killed US troops. In addition to reporting on the bounty offering, the story also claimed that the White House did nothing to respond to the threat.
"The Russia Bounty story is just another made up by Fake News tale that is told only to damage me and the Republican Party. The secret source probably does not even exist, just like the story itself. If the discredited @nytimes has a source, reveal it. Just another HOAX!" Mr Trump tweeted.
The New York Times attributed the reports to anonymous intelligence officials.
Mr Trump claimed that he was never briefed on the information.
"Do people still not understand that this is all a made up Fake News Media Hoax started to slander me & the Republican Party. I was never briefed because any info that they may have had did not rise to that level," Mr Trump said in another tweet.
The president's press secretary, Kayleigh McEnany, attempted to fend off claims that Mr Trump wasn't aware of the alleged threat to US troops because he chooses not to read his daily intelligence briefings.
"The president does read and he also consumes intelligence verbally," Ms McEnany said, when asked why Mr Trump wasn't reading his daily briefings. "The president, I will tell you, is the most informed person on planet earth when it comes to the threats we face."
Secretary of Defense Mark Esper said that the Pentagon had no evidence to corroborate the allegations made in the New York Times story.
Legislators briefed on the threat since the story broke have divided along party lines as to its credibility.
Texas Republican Mac Thornberry, ranking member of the House Armed Services Committee, did not call the reports a hoax but urged caution and further investigation, adding that US troops need to be protected.
"Every level of government needs to gather more information to understand this situation better," he said. "Measures have to be taken to be sure our troops are protected."
Mr Trump's political opponents have seized on the revelations to attack him ahead of the 2020 US election.
The Lincoln Project, a "Never Trump" conservative PAC that has endorsed Joe Biden for president, released a video ad in which a retired US Navy SEAL calls the president either a "coward" or "complicit" in the alleged Russia bounties.
"Any commander-in-chief with a spine would be stomping the living s*** out of some Russians right now, diplomatically, economically, or if necessary, with the sort of asymmetric warfare they're using to send our kids home in body bags," former Navy SEAL, Dan Barkhuff, said in the ad.
Mr Biden took his own shots at Mr Trump during a campaign stop in Wilmington, Delaware. He called the president's lack of response "an absolute dereliction of duty if any of this is even remotely true".
"I think the president has a lot to answer for, and we should get the answers quickly," Mr Biden said. "This president talks about cognitive capability. He doesn't seem to be cognitively aware of what's going on. He either reads and-or gets briefed on important issues, and he forgets it, or he doesn't think it's necessary that he needs to know it."
The former vice president suggested Mr Trump should have "at a minimum picked up the phone and said 'Vladimir, old buddy, if any is true ... you've got a big problem, a big problem'."
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