Trump tweeted, "Putin & I discussed forming an impenetrable Cyber Security unit so that election hacking, & many other negative things, will be guarded.."
He later added, "The fact that President Putin and I discussed a Cyber Security unit doesn't mean I think it can happen. It can't-but a ceasefire can,& did!"
Peskov confirmed the two leaders discussed the unit.
"There was, in fact, a discussion regarding the possibility to create such group. Time will tell whether it's going to be created or not. But the heads of the states discussed the possibility to create the group," Peskov told reporters Monday.
He continued: "Moreover, you know that the President gave Mr. Trump his statements and clarifications regarding the fact that any allegations about interference of some official Russian institutions, (and their connection) to cyberattacks during the US election is pure fiction. And in this context, they discussed that the best way of fighting against cyberterrorism is cooperation."
Peskov also said nothing was "promised" about the unit.
"You know, nobody promised each other anything. There was no such goal," he said.
When asked if Trump's tweet, which said that such a unit can't happen, affected Putin's trust for Trump, Peskov denied that this was the case.
"The positive thing is that the readiness to work in this direction was proclaimed," he said.
GOP lawmakers blast cybersecurity group
The President's initial announcement of the cybersecurity group provoked a strong reaction from members of his own party. GOP Sens. Ben Sasse, Lindsey Graham and John McCain spoke out in opposition to the idea.
Sasse tweeted on Sunday, "This obviously should not happen--& obviously will not happen. Why the President of the United States would tweet it is inexplicably bizarre."
And later added, "Putin is an enemy of free religion, speech, press and protest. He is now plotting future cyber attacks on America. He won't become a friend."
On Sunday, McCain told CBS's "Face the Nation," "I am sure that Vladimir Putin could be of enormous assistance in that effort since he's doing the hacking."
And Graham told NBC's "Meet the Press" that Trump's plan was "not the dumbest idea I've ever heard, but it's pretty close."
"(Trump is) literally the only person I know of who doesn't believe Russia attacked our election in 2016," he added.